[问答题]
$\text { The following diagram shows the graph of } y=x(\ln x)^{2}, x>0 \text {. }$
1. Given that the curve passes through the point (p, 0) , state the value of p .
The region R is enclosed by the curve ( $p \leq x \leq e $ ), the x -axis and the line x=e .
2. Integrate by parts twice to find the area of the region R .
Let $I_{n}=\int_{1}^{e} x^{2}(\ln x)^{n} \mathrm{~d} x, n \in \mathbb{N}$
3. a. Find the value of I_{0} .
b. Show that $I_{n}=\frac{1}{3}\left(e^{3}-n I_{n-1}\right)$.
c. Hence find the values of $ I_{1}$,$ I_{2}$ and $I_{3}$ .
The region R is rotated through $2 \pi$ radians about the x -axis.
4. Find the volume of the solid formed.
