Volumes by Slicing and Rotation about an Axis.

[tolove]

Homework Statement

Find the volume of the solid generated by revolving the region between the parabola x = y^2 + 1 and the line x = 3 about the line x = 3.

Homework Equations

The answer is found by integrating with respect to y with disk method, but I don't understand why my answer is incorrect when I try to integrate it with dx.

The Attempt at a Solution

The exact answer when integrated with respect to y is 64*pi*√(2)/15.

This is the integral I am incorrectly setting up somehow:
2*∫ pi * (√(x-1))^2 dx, x = 1 to 3

Any ideas why this doesn't work?

 

[Dick]

Homework Statement

Find the volume of the solid generated by revolving the region between the parabola x = y^2 + 1 and the line x = 3 about the line x = 3.

Homework Equations

The answer is found by integrating with respect to y with disk method, but I don't understand why my answer is incorrect when I try to integrate it with dx.

The Attempt at a Solution

The exact answer when integrated with respect to y is 64*pi*√(2)/15.

This is the integral I am incorrectly setting up somehow:
2*∫ pi * (√(x-1))^2 dx, x = 1 to 3

Any ideas why this doesn't work?

Integrating dy your cross sections are disks. If you want to integrate dx then your cross sections are shells. You need to use a different formula for the integral.

 

[tolove]

Integrating dy your cross sections are disks. If you want to integrate dx then your cross sections are shells. You need to use a different formula for the integral.

The cross-section of 3-d parabola isn't a circle?... ohhh, this isn't a 3d parabola, it's rotated.

Thank you very much!