The area A is defined as the region bounded by tw:gj)v8ui e;il bx7.dd g8:, dojp3m jhe curve y=$-\frac{x^{2}}{6}(x-6)$ and the x -axis for $0 \leq x \leq 6$ .
1. Sketch the curve on the diagram below, shading the area A .

2. Write down a definite integral that represents area A .
3. Find the area of A .
[Area A]=   units$^2$

  The area A is defined as the region bounded by the curve yj1r ,8ny p(g iy4d-jcy=$-\frac{x^{2}}{5}(2 x-5) $ and the x -axis for $ 0 \leq x \leq \frac{5}{2}$ .

1. Sketch the curve on the diagram below, shading the area A .

2. Write down a definite integral that represents area A .
Area A=$\int_{0}^{5 / 2}-\frac{x^{2}}{5}(2 x-5) \mathrm{d} x$ ;a=  ,b=  .
3. Find the area of A .
Area A=$\frac{a}{b}$ units$^2$ ; a=  ,b=  .

  The equation of a curve+3a9ww9 nfkw5nu pg*x is y=-$x^{3}+4 x^{2}+x-4$ . A section of the curve is shown on the diagram below, with the three x -intercepts labelled.

1. Find $\frac{\mathrm{d} y}{\mathrm{~d} x}$ = - ax$^2$+ bx+c ; a=  ,b=  ,c=  .
2. Write down the coordinates of the local maximum.
3. Write down an integral representing the area of the shaded region.
A=$\int_{1}^{4}\left(-x^{3}+a x^{2}+x-b\right) \mathrm{d} x$;a=  ,b=  .
4. Find the area of the shaded region.
A=$\frac{a}{b}$;a=  ,b=  .

  The equation of a curve is 6a dpl8qdnp04 y=-$x^{2}+8 x-12$ . A section of the curve is shown on the diagram below, with the two x -intercepts labelled.

1. Find $\frac{\mathrm{d} y}{\mathrm{~d} x}$ = -ax + b; a=  ,b=  .
2. Write down the coordinates of the local maximum, the local maximum is located at (  ,  )
3. Write down an integral representing the area of the shaded region.
4. Find the area of the shaded region.
A=$\frac{a}{b}$ units$^2$;a=  ,b=  .

  Let f(x)=$2 x e^{x} $ and g(x)=-2 f(x)-1 .
The graphs of f and g intersect at x=a and x=b , where a ＜ b

1. Find the value of a and of b .
2. Hence, find the area of the region enclosed by the graphs of f and g .
A=  .

  A reservoir of water has a cross-sectional arexozh irw 0 w0:dma68j)a shown in the shaded region of the diagram below, ha06o0: rmxw zijdw8)where x and y values are in metres. The bottom of the reservoir (the curve shown) has an equation in the form

$y=k(x-8)^{2}$

1. Find the area of the rectangle OMNP, The area of the rectangle OMNP is determined by A$_1$=   m$^2$.

2. Determine the value of k is $\frac{a}{b}$;a=  ,b=  .
3. Find the cross-sectional area of the reservoir.
The number is    m$^2$.

  A section of the cure;zu xoj9y 2x47:wwig ve with equation y=2(x-3)(x+8) is shown.

1. Write an integral for the shaded area A.
[Area A] = $\int_{a}^{b} 2(x-3)(x+8) \mathrm{d} x$; a=  ,b=  .
2. Find the area of A is    units$^2$ .
3. Find the area of B is    units$^2$ .

  The following diagram shows the parabolic shape of a gh o(mf3 z(q fb9ievl.9zxv54dateway arch that has a span of 12 metres and a maximum height of 8 metqf4f3 d(blx5hm.9z9 eo vivz( res.

The curve has an equation in the form y=$k(x-6)^{2}+8$ .
1. Determine the value of k is $\frac{a}{b}$;a=  ,b=   .
2. Write down an integral that represents the cross sectional area under the arch shown as OMN.
A=$\int_{0}^{12}\left[-\frac{a}{b}(x-6)^{2}+c\right] \mathrm{d} x$;a=  ,b=  ,c=  
3. Find the cross sectional area under the arch.Evaluating the integral in part (b), we get a=   m$^2$.

  Let f(x)=$9-2 \ln \left(x^{2}+4\right) $, for $x \in \mathbb{R}$. The graph of f passes through the point (p, 3) , where p>0 .
1. Find the value of p is   .

The following diagram shows part of the graph of f .

The region enclosed by the graph of f , the x -axis and the lines x=-p and x=p is rotated $360^{\circ}$ about the x -axis.
2. Find the volume of the solid formed is   .

  Consider the region enclosed by np6g1 vki2ymw*/b 4. jlmi 4lethe graph of the function f(x)=$e^{x / 2}$ , the x -axis and the vertical lines x=0 and x=3 .
1. Sketch the function f and shade in the region.

2. 1. Write down a definite integral that represents the shaded region, it is $\text { 1. } \int_{a}^{3} e^{x / b} \mathrm{~d} x \backslash$,a=  ,b=  .
2. Calculate the area of the shaded region is   .
3. Use the trapezoidal rule with n=6 trapeziums to estimate the area of the region.

  Let f(x)=$\frac{1}{\sqrt{3 x-6}}$ , for x>2
1. Find $\int(f(x))^{2} \mathrm{~d} x$ = $\frac{1}{a} \ln (b x-6)$+C;a=  ,b=  .

Part of the graph of f is shown in the following diagram.

The shaded region R is enclosed by the graph of f , the x -axis, and the lines x=3 and x=11 .
2. Find the exact volume of the solid formed when R is rotated $360^{\circ}$ about the 3 x -axis.
V= $\frac{\pi}{a}$$ln_{b}$,a=  ,b=  .

  On the Persian Gulf, there is a small peninsula that can be mode iq x.7*:vt jva.4hzuxlled by the equation *a.7xxv jq:hut. ivz4 y=$-0.2 x^{3}+1.2 x^{2}-x$ where x and y are measured in kilometres, as shown in the following diagram. Points $\mathrm{P}$ and $\mathrm{Q}$ are on the coastline.

1. Find the coordinates of $\mathrm{P}$ is (  ,  ) and $\mathrm{Q}$ is (  ,  ) .
2. Find the furthest distance that the land protrudes from the coastline.
3 . Find the area of the peninsula，The area of the peninsula is given by A=   km$^2$.

  The graphs of y=-0.5 x+2.5 2tl5mcd eep) bt4-upf p cv93+ and $y=0.5 x^{2}+0.5 x+1$ intersect at (-3,4) and (1,2) , as shown in the following diagrams.

1. Calculate the shaded area in diagram 1 , the area enclosed by the lines y=-0.5 x+2.5 x=-3, x=1 and the x -axis, A$_1$=   units$^2$
2. 1. Write down an integral for the shaded area in diagram 2 , the area enclosed by the curve y=$0.5 x^{2}+0.5 x+1$ , the lines x=-3 and x=1 , and the x -axis.
A$_2$=$\int_{-3}^{1}\left(0.5 x^{2}+0.5 x+a\right) \mathrm{d} x$;a=  .
2. Calculate the area of this shaded region,A$_2$≈   units$^2$
3. Hence, determine the area enclosed by the line y=-0.5 x+2.5 and the curve y= $0.5 x^{2}+0.5 x+1$

  A curve y=f(x) has a gradient func;qqi2wuu a/ qi5q:j-bhxi57psynl)/ tion of f^{\prime}(x)=a x+1 . The diagram below shows part of the curve, with the x and y intercepts labelled.

1. Find the equation of the curve is y=-ax$^2$+bx+c; a=  ,b=  ,c=  .
2. Find the area under the curve from 0A=$\frac{a}{b}$units$^2$,a=  ,b=  .

  A curve y=f(x) has a gradient flkduim1 hbcjju1a(g33) qx +/unction of $f^{\prime}(x)=a x+5 $. The diagram below shows part of the curve, with the x and y intercepts labelled.

1. Find the equation of the curve,y=$-ax^2+bx+c$;a=  ,b=  ,c=  .
2. Find the area under the curve from 0 ,b=  .

  The shape of a bowl is given by revolving thejgyx lcp.tax/ 6v 39x: curve y=$\frac{1}{3} x^{\frac{7}{4}}$, $0 \leq x \leq 6 through 360^{\circ}$ about the y -axis. The units are in centimetres.

1. Find the height of the bowl.
h≈   cm.(Two decimal places)
The bowl is completely filled with water.
2. Find the volume of water in the bowl.
V≈   cm$^3$(round number)

  The following diagram sh dz-:5g wxsrd,ows the graph of the function $f(x)=\frac{b}{a} \sqrt{a^{2}-x^{2}}$

Find an expression, in terms of a and b , for the volume of the solid formed by rotating the curve $360^{\circ} $ about the x -axis.
we get V=$\frac{x{\pi}ab^2}{y}$;x=  ,y=  .

  The following table shows the rate, R , at which rubbish is produceg/(f; -3kbl2 nw yyzsrd by a town, in tonnes per day. t represents ykgsn; zbwlry-(f3 /2 the time in days since the rate of rubbish production was first recorded.

The total amount of rubbish produced by the town over the 3-day period is equal to the area underneath the graph of the curve R(t) .

1. Use the trapezoidal rule with 6 intervals to estimate the total amount of rubbish produced by the town over the 3-day period；The total amount of rubbish produced is approximately    tonnes (round number please)

The actual amount of rubbish produced over the 3 days was 191 tonnes.
2. Find the percentage error between the estimate in part (a) and the actual amount of rubbish produced;The percentage error between the estimate in part (a) and the exact value can be found using the percentage error formula is close to   % (tow decimal please)

  A curve y=f(x) passes thr dc sg)9n/p*/ 6s6/ujz f1tila8gq d stw+(bciough point $\mathrm{P}$(2,7) and has a gradient of $ f^{\prime}(x)= 4 x-3 $.
1. Find the gradient of the curve at point P ;f(x)=  .
2. Find the equation of the tangent to the curve at point P; y=kx-b,k=  ,b=  .
3. Determine the equation of the curve; the equation of the curve is y=ax$^2$-bx+c;a=  ,b=  ,c=  .

  The following diagram zt;1np2 lf(zyshows part of the graph of f(x)=\left(16-x^{2}\right)(x+ 3 ), x \in \mathbb{R} . The shaded region R is bounded by the x -axis, y -axis and the graph of f .

1. Write down an integral for the area of region R ;[Area R] = $\int_{0}^{4}\left(a-x^{2}\right)(x+b) \mathrm{d} x$;a=  ,b=  .
2. Find the area of region R ; [Area R]=    units$^2$

A trapezoid A B C D has four vertices with the coordinates shown as below.

3. Find the value of a , the x -coordinate of point $\mathrm{D}$ , such that the area of the trapezoid is equal to the area of region R .
Since the area of the trapezoid ABCD is equal to the area of region R, we get a is   .

  The following diagram shows part ofvxz(0rmv.. m( oudq cx8f4;pw the graph of f(x)=$x^{3}-3 x^{2}-4 x+ 12$, $x \in \mathbb{R}$ . The shaded region R is bounded by the x -axis, y -axis and the graph of f .

1. Write down an integral for the area of region R ；[Area A]=$\int_{-2}^{0}\left(x^{3}-a x^{2}-4 x+b\right) \mathrm{d} x$;a=  ,b=  
2. Find the area of region R is    units$^2$

A parallelogram A B C D has four vertices with the coordinates shown as below.

3. Find the value of a , the y -coordinate of points $\mathrm{B} $ and $ \mathrm{C}$ , such that the area of the parallelogram is equal to the area of region R . a=  .

  A company is going to make4 x5,4gphl gjy an excavation to access a mine. The cost of excavating 5,pxhg 4lj4g yis \$ 12 per $\mathrm{m}^{3}$ . The following table shows an estimate of the areas of horizontal crosssections, at 5 -metre intervals, of the underground to be removed where the height is measured in metres below the ground level.


1. Use the trapezoidal rule to find an estimate of the volume of ground to be removed，we obtain the V ≈   m$^3$ (round number)
2. Hence, or otherwise, estimate the total cost of excavation;The total cost of excavation is $\$$   .
​

  A piece of jewellery is bei s )f;u j+ocpv/hilwt , t)5jqj+dy8s6ng designed for an auction. Its shape is based on the graph 8/y j)tsjf5ov6 tclid+wqu +h);,psjof

$y=\frac{x}{50} \sqrt{625-x^{2}} \quad \text { for } 0 \leq x \leq 25$

which is revolved $360^{\circ}$ about the x -axis, where x and y are measured in $\mathrm{mm}$ .

1. Find the area bounded by the graph and the x -axis.
Area ≈   .
2. Hence or otherwise, find the volume of the jewel is(≈)   mm$^3$.
3. If $1 \mathrm{~cm}^{3}$ of silver weighs 10.5 grams, and 1 gram of silver costs $\$ 1.50$ , find the cost of the silver used for the jewel to the nearest dollar is(≈)   .

  Rain is filling an empty wat wpg o0.n0wq c,fug67zer reservoir at a rate of r litres per minute, where r=ww uf06zp g,0gc 7.noq$\frac{t}{5}$ and t is the time measured in minutes. The graph of this is shown below.

1. Find how much water has filled the reservoir after 1 hour ; we get V=    L
2. Determine an expression for the amount of water that has filled the reservoir for any given number of minutes, m ;The volume of water that has filled the reservoir after m minutes can be expressed as V=$\frac{a}{b}$m$^2$ (L);a=  ,b=  .

  The following diagram shwwh7;-ot/d6 sx*jfn uows the graph of f(x)=$\frac{4 x}{x^{2}+1}$ , for $0 \leq x \leq 6$ , and the line x=6 .

Let R be the region enclosed by the graph of f , the x -axis and the line x=6 .
1. Show that $\int f(x) \mathrm{d} x=2 \ln \left(x^{2}+1\right)+C$ .
2. The exact value for the area of the region R is $\ln k$ , where k is a positive integer. Find the value of k is   .

  The diagram below shows the graphs of funczcz8um oev 9(:tions $y=4-2 x^{2}$ and $y=2 x^{3} $.

1. Find the volume resulting from a rotation of the shaded region shown in the diagram through $2 \pi $ about the x -axis.
V≈   .( one decimal pleaces)
2. Find the volume resulting from a rotation of the region shown in the diagram through $2 \pi $ about the y -axis.V≈   .( two decimal pleaces)

  The diagram below shows the gr(n5 )azkv)ege:oaa ex . mnf79aphs of functions $y=5-3 x^{3}$ and $y=2 x^{4}$ .

1. Find the volume resulting from a rotation of the region shown in the diagram through $2 \pi$ about the x -axis.V≈   (one decimal pleaces)
2. Find the volume resulting from a rotation of the region shown in the diagram through $2 \pi $ about the y -axis.V≈   (two decimal pleaces)

  Bellatrix Aviation manufactures business jets. The profit the company makes b:fy0w5 ifientu5x 68, P , in millions of dollars, :5fin5fi wxtu 60 eyb8changes based on the number of jets, x , they manufacture per quarter.
The rate of change of the profit made from manufacturing x jets is modelled by

$\frac{\mathrm{d} P}{\mathrm{~d} x}=-0.4 x+16, \quad x \geq 0 $.


In the previous quarter, the company made a profit of 165 million dollars from manufacturing 20 jets.
1. Find an expression for P in terms of x .Therefore we obtain P= $-{a}x^2 + b x - c$;a=  ,b=  ,c=  .

The company has the capacity to increase the number of jets they manufacture.
2. Describe how the profit changes if the company increases manufacturing to over 40 jets per quarter and up to 75 jets per quarter.

  The water in a pool is being pumped out iz jd2o+* f+ktunto a nearby lake at a rate of r litres per second where +jdo*+kufz2 tr=$\frac{2 t}{275}$ and t is time measured in seconds. The graph of this is shown below.

1. Find out how much water has been pumped out after 1 minute,We get V≈   L
2. Determine an expression for the volume of water that has been pumped out of the pool for any given number of seconds, s ;The volume of water that has been pumped out of the pool after s seconds can be expressed as
v=$\frac{a}{b}$s$^2$ (L) ;a=  ,b=  .

  Consider the region R enclosey;s 6l8xiy+ r-t qj; bp14fhhqd by the curve $y=-3(x-2)(x-8)$ and the x axis.

1. Sketch the curve and shade the region R on the following axes.

2. 1. Write down an integral that represents the area of region R ;The area of region
R is determined by A=$\int_{2}^{8}-a(x-2)(x-b) \mathrm{d} x$ ;a=  ,b=  .
2. Find the area of region R is    units$^2$.
3. Estimate the area of region R using five trapezoids.
4. Find the percentage error between the exact value found in part (b)(ii) and the estimation in part (c);The percentage error between the estimation in part (c) and the exact value in part (b)(ii) can be found using the percentage error formula [% error]=   %

  Let f(x)=$x(x-3)^{2}$ , for $0 \leq x \leq 4$ . The following graph shows f .

Let R be the region enclosed by f , the x -axis and the lines x=0, x=3 .
1. Show that $\int f(x) \mathrm{d} x$=$\frac{x^{4}}{4}-2 x^{3}+\frac{9 x^{2}}{2}+C$.
2. Find the volume formed when R is rotated $360^{\circ}$ about the x -axis ;The volume is given by the integral V=   .
3. The diagram below shows part of the graph of the quadric function g(x)=x(a-x) . The graph of g crosses the x -axis when x=a .

The area enclosed by the graph of g is equal to the area of R . Find the value of a is   .

  BioNature Inc. is considering the monthly cost of producing straws made froj8qm9 :mphib7vug r *.m paper. It is estimated that the rate of change in monthly cost:pvmi.7jgbmu* 9r8qh with respect to the number of paper straws produced is modelled by

$C^{\prime}(q)=0.006 q^{2}-0.48 q-12, \text { for } q>0$

Where the monthly cost, $ \$ C$ , is in thousands and the number of straws, q , is also in thousands.
1. Find the number of paper straws that the company should produce in a month to minimize the cost.

The company produces 110000 paper straws with a cost of $\$ 88000$ in a certain month.
2. Find an expression for C in terms of q ;we obtain C(q)=$0.002 q^{3}-aq^{2}-12 q+b$;a=  ,b=  

  A healthcare company is designing a new deodorant sti) c y9xtv/a m:2j(erz4 mi:rhick that can be modelled by the shape of a parabola as the top, vertical lines as the sides and a horizontal line as the base, on the x -axis. The param:jv)x ry( et2/ci4ar zmh 9i:bola has end points at (0,7) and (5,7) , and vertex at (2.5,8) .
This design is shown in the diagram below. All distances are measured in centimetres.

The quadratic curve can be expressed in the form y=$a x^{2}+b x+c$ for $0 \leq x \leq 5$ .
1. 1. Find the value of c is  .
2. Using the points (2.5,8) and (5,7) , write two equations in a and b .
3. Hence, find the equation of the quadratic curve; we get y=-ax$^2$+bx+c;a=  ,b=  ,c=  .
2. Find the area of the shaded region in the deodorant design.

  Water is pumped from a river basin into an irrigated field at a rate of r=iw1x +lbgvv9hopzulbw/.;lg4/3j f1+3y o qw \fraclxy 4 1o33i+jbwz lbq;plvvfg /1. w/wu+9 hgo{t^{2}}{25} where r is measured in cubic metres of water per minute and t is the time measured in minutes. The graph of this is shown below.


1. Write down an integral that represents the area of the shaded region. It is A =$\int_{0}^{60} \frac{t^{2}}{a} \mathrm{~d} t$ ;a=  .
2. 1. Find the area of the shaded region is    m$^2$.
2. Interpret the meaning of the shaded area in the context of this question.
3. Find the number of minutes it takes to pump 500 $\mathrm{~m}^{3}$ of water into the field.
m≈   min (one deciaml pleaces).

  The following diagram shows a section of a curve y=f(x) and the coordinates onb.n r y aher21kk42rr3y730hh rb sk;f six points that lie onbn3 ran201bresyk3hr r k4h.2k y7h r; the curve.


1. Estimate the area of the shaded region, giving your answer to two decimal places.
$\int_{1}^{5} y \mathrm{~d} x$ $\approx$    units^2
The equation of the curve was found to be y=2 $x^{2}-4 x+20$ .
2. Write an integral that represents the area of the shaded region.
A= $\int_{1}^{5}\left(a x^{2}-b x+c\right) \mathrm{d} x$; a=  ,b=  ,c=  .
3 . Find the area of the shaded region, giving your answer in exact values.
A=$\frac{a}{b}$ $units^2$ ; a=  ,b=  .
4. Find the percentage error between the estimation in part (a) and the exact value found in part (c).

  The diagram below shows part of the grap /o-nkw9c f33gnd 7lcutq5r-jhs of y=$(x+2)^{2}$ and y=$(6-x)^{2}$ .

1. Find the coordinates of the points P, Q and R .
P is (  ,  ) ;Q is (  ,   ) ;R is (  ,  )
The shaded region is rotated through $360^{\circ}$ about the x -axis.
2. Calculate the volume of the solid obtained.

  Consider the region R enclosed by the curve y qcosr-e8+5hd0 9lemq=$\frac{1}{2}(x+2)(x-3)$ and the x -axis.
1. Sketch the curve and shade the region R on the following axes.


2. 1. Write down an integral that represents the area of region R .
2. Find the area of region R is A ≈   units$^2$(one decimal please)
3. Estimate the area of region R using five trapezoids.
it is ≈    $units^2$ (round number)
4. Find the percentage error between the exact value found in part (b)(ii) and the estimation in part (c).

  In a memory competition, the rate at which English speaking- bsju561 gpt1ydq h8p participants memorise French vocabulary can be modelled by the eq8sj6guh1 t1pbq- ydp5uation

$\frac{d M}{d t}=1-0.01 t$, $\quad t \geq 0$,

where M(t) is the number of words memorised in t minutes.
The number of words participants have memorised at the beginning of the competition, t=0 , is zero.
1. Find the equation for the number of memorised words at time t . M(t)= t - a$^{-3}$ t$^2$ ; a=  .

The competition ends when students get so tired that the number of additional memorised words per minute becomes zero.
2. Find the value of t when the competition will end.
t=   minutes.
3. Determine
1. the domain of M(t) is [  ,  ],
2. the range of M(t) is [  ,  ].

  Kai launches stones from a slingshot into the altbqi5;t(ex + ir. Each time he launches a stone, he records the angle, \theta , in degrees, at which the stone is launched, and the horizontal distance, L , in metres, which the stone tr;x t+ql(5tbe iavels from his shooting position to where it first lands. The diagram below illustrates how the stones travel in the air when launched at different angles.


Kai analyses his results and concludes:

$\frac{\mathrm{d} L}{\mathrm{~d} \theta}=-0.08 \theta+3.3, \quad 0^{\circ} \leq \theta \leq 90^{\circ}$ .

1. Determine whether the graph of L versus \theta is  ----待选择----increasing decreasing  at $\theta= 50^{\circ}$ .

Kai observes that when the angle is $30^{\circ} $, the sling stone will travel a horizontal distance of $90 \mathrm{~m}$ .
2. Find an expression for L in terms of $\theta$ .
L= -0.04$\theta$$^2$+ a$\theta$+b ;a=  ,b=  .

  The cross-sectional view of a two-lane road tunnel system is sh x )0dht4i/peuown on the axes below. The left and right lane tunnels are separated by a 2 metre thick concrete wall) pituxd/eh0 4. The right-hand tunnel passes through the points $\mathrm{A}$, $\mathrm{B}$, $\mathrm{C}$ and $\mathrm{D}$ and its height, in metres, above the base of the tunnel, is modelled by f(x)=-$0.04 x^{3}$+ 0.41 $x^{2}$, $4 \leq x \leq 10$ , relative to an origin $\mathrm{O} $.


Point A has coordinates (4,4) and point $\mathrm{D}$ has coordinates (10,1) .
1. Find the height of the right-hand tunnel when:
1. x=6 ; f(6)=   (m)
2. x=8 ; f(8)=   (m)
The left-hand tunnel can be modelled by a function g(x) , found by reflecting f(x) in the line x=3 .

2. Find the equation of g(x) .g(x)=$0.04 x^{3}-a x^{2}-bx+6.12$;a=  ,b=  .

3. 1. Find $g^{\prime}(x)$ .$g^{\prime}(x)=ax^{2}-b x-0.6$;a=  ,b=  .
2. Hence find the maximum height of the left-hand tunnel.
4. 1. Write down an integral which can be used to find the cross-sectional area of the left-hand tunnel.
2. Hence find the combined cross-sectional area of both tunnels.

  A fire truck is pumping water at a ratep 8anjjc: cg+698ro bt of r=$\frac{2 t^{2}}{35}$ where r is in litres per second and is the time measured in seconds.
The graph of r is shown below.

1. Write down an integral that represents the area of the shaded region.
$A=\int_{x}^{y} \frac{a t^{2}}{b} \mathrm{~d} t$;a=  ,b=  ; x=  ,y=  .
2. 1. Find the area of the shaded region, rounding your answer to the nearest whole number.
2. Interpret the meaning of the shaded area in the context of this question.
3. Calculate the number of seconds it takes for the fire truck to pump $1000 \mathrm{~L}$ of water.
s≈   s (one decimal plaeaces)

  The diagram below shows the crofzk gwtau n:/9* 6i.shss-sectional area of a mound of beach sand created :/ tw6n*sfa. izkg h9uafter a high tide.


The curve of the cross section can be modelled by the following equation

$y=\frac{x^{2}(90-x)}{1800}$

where y represents the vertical height of the mound in $\mathrm{cm}$ and x denotes the horizontal width in $\mathrm{cm}$ , from the start of the mound.
1. At a horizontal width of x=30 , determine
1. The vertical height of the mound at this point; y=   cm.
2. The gradient of the mound curve at this point. y'=$\frac{a}{b}$;a=  ,b=  .
2. 1. Find the value of x which corresponds to the maximum the vertical height of the mound;it is   .

2. Hence, find the maximum vertical height of the mound. y=   cm
3. Calculate the cross-sectional area of the mound, rounding your answer to one decimal place.

A child uses a toy shovel to remove the top of sand mound, as illustrated by the line segment $\mathrm{MN}$ below. Point M has coordinates at (30,30) .

4. Determine the coordinates of point N .

The cross-sectional area removed by the child can be expressed by the following integral

$\int_{p}^{q} \frac{x^{2}(90-x)}{1800} \mathrm{~d} x-R$,

5. Determine the values of p, q and R . p=  ; q=  , R=  .

  The following table shows the x and y coordinates of five poi*1z zzl+ yk,)y 4xsv6iofqdupv--3 dents that lie on a curve y=f(x) .vy,z veku op6yxq3-df)-14s lzdz+i *


1. Estimate the area under the curve over the interval 2
The equation of the curve was found to be y=$x^{3}-7 x^{2}+14 x-2$ ;it is Approximately equal to    units$^2$( one decimal please)
2. 1. Write down an integral that represents the area under the curve over the interval.
2. Find the exact value of the area under the curve over the interval.A=$\frac{a}{b}$;a=  ,b=  .
3. Find the percentage error between the estimation in part (a) and the exact value in part (b). Provide a reason for the difference.

  The following table shows the x and y coordinates of fivtd*33ck6st4wwhv+s* 9h v mk oe points that lie on a curve y=hwo4sc w6v3ksht+3 * dvk*9m tf(x) .

1. Estimate the area under the curve over the interval 1
The equation of the curve was found to be y=-$\frac{1}{10}(4 x+5)(x-8)$ .
it is approximately equal to    units$^2$

2. Find the exact value of the area under the curve over the interval 1A=$\frac{a}{b}$;a=  ,b=  .
3. Find the percentage error between the estimation in part (a) and the exact value in part (b). Provide a reason for the difference.it is approximately equal to    %(two decimal places)

  During a trip to a fore s4,bimv5ab4**zv cktst to forage for mushrooms, Viviane finds a giant mushroom. She decides to model t 4k acvbzvb**m4i,ts 5he shape of the mushroom to find its volume.

After taking a photo of the mushroom and zooming in to get its real size, she rotates the photograph and estimates that the cross-section passes through the points (0,3) (15,3),(15,15),(22,10),(23.5,6) and (24,0) , where all measurements are in centimetres. The cross-section is symmetrical about the x -axis, as shown below.

Viviane models the section from (0,3) and (15,3) with a straight line.
1. Write down the equation of the line that passes through these points.
y=  .
Next, Viviane models the section that passes through the points (15,15),(22,10) , (23.5,6) and (24,0) with a quadratic curve. it is (  ,  )
2. 1. Use your G.D.C. to find the equation of this quadratic curve.
y≈-1.82(x-a)$^2$+a;a=  .
2. By considering the gradient of the curve at the point (15,15) , explain why this may not be a good model.

Viviane thinks she can obtain a better model if a quadratic passing through the point (24,0) with a maximum point at (15,15) is used.
3. Find the equation of this model, in the form y=a(x-h)^{2}+k .

Using this new model, Viviane proceeds to estimate the volume of the mushroom by finding the volume of revolution about the x -axis.
4. 1. Write down an expression for her estimate as a sum of two integrals.
2. Find the volume of the mushroom estimated by Viviane.
V≈   cm$^2$ (round number)

  Lohan receives an antique vase as a gift from her grandparent8bl tt+ia *s2xs. She decides to model the shap* i2tstla8bx +e of the vase to calculate its volume.

She places the vase horizontally on a piece of paper and uses a pencil to mark five representative points (0,5),(7.5,10),(17,7.5),(25,3) and (35,7.5) , as shown below. These points are connected to form a symmetrical cross-section about the x axis. All units are in centimetres.


Lohan initially uses a straight line to model the section from (25,3) to (35,7.5) .
1. Determine the equation of the line that passes through these two points.
y=ax-b;a=  ,b=  .
Lohan thinks that a quadratic curve might be a good model for the section from (0,5) to (25,3) . She carries out a least squares regression using this model for the points she has recorded.
2. 1. Determine the equation of the least squares regression quadratic curve found by Lohan.
2. By considering the gradient of the curve at the point where x=10 , determine whether the quadratic regression curve is a good model or not.

Lohan decides that a cubic curve for the entire section from (0,5) to (35,7.5) would be a better fit.

3. Find the equation of the cubic model.

Using this model, Lohan estimates the volume of the vase by calculating the volume of revolution about the x -axis.
4. Find the volume of the vase estimated by Lohan.

Lohan subsequently fills the vase with water and discovers that the true volume is 5500 $\mathrm{~cm}^{\wedge} $3 .
5. Calculate the percentage error in Lohan's estimate of the volume.
Using the formula for percentage error, we have
ϵ=  %
​

  A function is given by m8cm n)4-6ypy p8acej $f(x)=4 x^{3}-x for -\frac{1}{2} \leq x \leq \frac{1}{2}$ .
1. Sketch the graph of f for the given domain, labelling all intercepts and stationary points.

2. Find an approximation for $I=$\int_{0}^{1 / 2} f(x) \mathrm{d} x$$ using the trapezoidal rule with a strip width of 0.1 .
3. Find the exact value of I in part (b).The exact value of I is -  
4. Explain why the approximation found in part (b) is larger than the exact value found in part (c).
5. Write down the value of $J=\int_{-1 / 2}^{1 / 2} f(x) \mathrm{d} x$ without calculation. Give a reason for your answer.
I=  .

  The table below shows th(t hl7 t-dxkvh x*x- z9smk1f8e daily revenue, R , in thousands of USD, of a toy store during the month 8(k -vf1lztmk xhx7d-9 txs*hof December. The first row of the table shows the day of the month (t ) and the second row the revenue of that day in thousands of USD.

1. 1. Generate a scatter diagram for this data on the axes below.

2. Identify a possible outlier and give a possible reason for this outlier;
3. Give a reason why plotting R vs $ \log (t)$ is likely to be the best linearisation method.
2. Linearise the data and determine an appropriate linear model for the daily revenue R in terms of day of the month in December, t .
3. Using your model, predict the revenue after 3 weeks in December.
4. The total revenue for the month of December can be estimated by $ \int_{a}^{b} R \mathrm{dt}$ .
1. Write down the values of a and b .a=  ,b=  .
2. Estimate the total revenue in December using your model.
TR≈    thousads of USD

  A company that manufactures and sells cardboard boxes has a box with an open-to*f c9hu e;+r*yz zkfg5p design. This box is constructed from a rectangular cardboard she*khfeufgry zc5 ;9z *+et with a length of 2 meters and a width of 1.2 meters, as illustrated in the diagram below. The box is formed by cutting squares of equal side length ( x meters) from each corner and folding up the sides.

1. Show that the volume of the box can be described by the function $V(x)= 4 x^{3}-6.4 x^{2}+2.4 x $.

2. 1. Find $V^{\prime}(x)$= ax$^2$-bx+c;a=  ,b=  .c=  .
2. Hence or otherwise, find the value for x that maximises the volume of the box;
x=   m
3. Hence, find the maximum volume of the box.

3. Sketch the graph of V(x) on the axes below for the domain 0
4. 1. Write down an integral representing the area between the graph of V(x) and the x -axis;
2. Hence, find this area between the graph of V(x) and the x -axis.

Let A(x) be a function representing the outside surface area of the box.
5. Determine the function A(x) .
A(x)=-ax$^2$+b;a=  ,b=  .
6. Given that the volume of the box is maximised, find the outside surface area of the box.

  All lengths in this question are packv201sge ( in metres.
Consider the function $f(x)=\sqrt{\frac{16-4 x^{2}}{7}}$ , for $-2 \leq x \leq 2 $. In the following diagram, the shaded region is enclosed by the graph of f and the x -axis.

A rainwater collection tank can be modelled by revolving this region by $360^{\circ}$ about the x -axis.
1. Find the volume of the tank.
V=   m$^3$
Rainwater in the tank is used for drinking, cooking, bathing and other needs during the week.
The volume of rainwater in the tank is given by the function g(t) , for $ 0 \leq t \leq 7$ , where t is measured in days and g(t) is measured in $\mathrm{m}^{3}$ . The rate of change of the volume of rainwater in the tank is given by $g^{\prime}(t)=1.5-4 \cos \left(0.12 t^{2}\right)$ .
2. The volume of rainwater in the tank is increasing only when a 1. Find the value of a and the value of b .
2. During the interval a d=   m$^3$
When t=0 , the volume of rainwater in the tank is 8.2 $\mathrm{~m}^{3}$ . It is known that the tank is never completely full of rainwater during the 7 day period.
3. Find the minimum volume of empty space in the tank during the 7 day period.

V$_{min}$=  m$^3$

  Consider the function defiv0tp+ -:7sijm/*t g/rtei tw jned by $f(x)=(1-x) \sqrt{2 x-x^{2}} $ where $0 \leq x \leq 2 $.
1. Show that $ f(1-x)=-f(1+x)$ , for $-1 \leq x \leq 1$ .
2. Find $f^{\prime}(x) $=$\frac{2 x^{2}-a x+b}{\sqrt{2 x-x^{2}}}$;a=   ,b=  .
3. Find the x -coordinates of any local minimum or maximum points.

4. Find the range of f is [-   ,   ]
5. Sketch the graph of y=f(x) , indicating clearly the coordinates of the x -intercepts and any local maximum or minimum points.
6 . Find the area of the region enclosed by the graph of y=f(x) on the x -axis, for $ 0 \leq x \leq 1$ .

  Water is flowing out e/aieynp65z10g5py w r-y el.of a tank at a rate modelled by the function

$R^{\prime}(t)=4 \sin \left(\frac{t}{100}\right)$


Water is flowing into the same tank at a rate modelled by the function

$S^{\prime}(t)=\frac{12 t^{2}}{1+t^{3}}$


Both R^{\prime} and S^{\prime} are measured in $\mathrm{m}^{3}$ , and t in hours for $ 0 \leq t \leq 10$ .
1. Find the interval on which the amount of water in the tank is increasing.

2. Find an expression, T , for the amount of water in the tank at time t if initially there was 25 $\mathrm{~m}^{3}$ of water in the tank.
T(t)=4$\ln \left(t^{3}+1\right)$+400 $\cos \left(\frac{t}{100}\right)$-a; a=  .
3. Hence, or otherwise, find the value of the maximum amount of water in the tank and the time it occurs.