1. ## Volume

Find the volume of the solid obtained by rotating the region bounded by the given curves about the specified line. y=2x^2, 0<x<2, y=8; about the y-axis.

2. Originally Posted by asnxbbyx113
Find the volume of the solid obtained by rotating the region bounded by the given curves about the specified line. y=2x^2, 0<x<2, y=8; about the y-axis.
The area of a disk of radius x=sqrt(y/2) is pi x^2= pi y/2. The volume you
seek is the integral of this area from y=0 to y=8 or:

V = integral(y=0:8) pi y/2 dy = pi/2 integral(y=0:8) y dy = pi/2 [8^2/2 - 0] = 16 pi

RonL

3. Hello, asnxbbyx113!

This can be done with cylindrical shells, too.

Find the volume of the solid obtained by rotating the region bounded by:
. . y = 2x², 0 < x < 2, y = 8; about the y-axis
Code:
          |
8+ - - - - *
|:::::::::
|::::::::*
|::::::*
|:::*
----*----------+---
|          2
. . V . = . x·y dx

. . . . . . . . . . . . . . .
2
We have: .V .= .
x(8 - 2x²) dx . = . 16π
. . . . . . . . . . . . . .
0