Volume rotation problem-help!

keta

Hi, can someone please explain how to do this problem (I get confused on how to set up the integral to find the volume when the text asks for volume about a line other than an axis):

Find the volume of the solid generated by revolving the region enclosed by the parabola $y^2=4x$ and the line $y=x$ about the line $x=4$ .

II. Find the volume when the region is rotated instead about the line $y=4$ .

Thanks.

**Posted:** Tue Feb 08, 2005 4:22 pm

kunny

Aprroach to 1

First of all,you cut this solid by the plane $x=t\ (0\leqq t\leqq 4)$ .
Let $P(t,2\sqrt{t}),Q(t,t))$ , the desired volume is equal to the volume of rotating segment $PQ$ around $x$ -axis.That is to say,you can find the volume by subtracting the volume of cone with the radius of base 4,hight4 from that of the solid generated by rotating the region enclosed by the parabolla $y=2\sqrt{x}$ and the line $y=x$ .

Therefore the desired volume is as follows.

$\int^4_0 \pi (2\sqrt{t})^2 dt-\frac{1}{3}\pi\cdot 4^2\cdot 4$

kunny

**Posted:** Tue Feb 08, 2005 5:03 pm