Volume of Solid Between Curves: Region, Sketch, and Solution

[phillyolly]

Homework Statement

Find the volume of the solid obtained by rotating the region bounded by the given curves about the specified line. Sketch the region, the solid, and a typical disk or washer.

Homework Equations

The Attempt at a Solution

Sometimes, I struggled with drawing a solid and a typical disc. Also, I am unsure with the solution for problem 4. Do you see any mistakes?

I would so much appreciate your help.

 

[phillyolly]

This is a rare time when no one replied. Sad.

 

[Mark44]

You'll probably do better submitting one problem at a time.

The first couple of problems are missing some information - namely, the line around which the region is rotated. For example, in #2, you have the region bounded by y = 1 - x² and y = 0, but you didn't say what this region is rotated around.

 

[Bohrok]

It's probably rotated around y = 0, especially since the problem mentions "disk and washer."

I'm guessing the cylinder shape you drew was supposed to be a typical disk/washer the problem asked for, but that's not what you want. When the region is rotated around a horizontal line, disks and washers will be vertical, between the axis of revolution and the function for disks or between the two given functions for washers.

On #4, what are the last three equations for?
y = 0, x = 2, x = 4

 

[Mark44]

It's probably rotated around y = 0, especially since the problem mentions "disk and washer."

Disks and washers can be oriented either vertically or horizontally.