# Find the volume of the described solid.

• Oct 13th 2009, 08:00 PM
MarkOfEternity
Find the volume of the described solid.
The base of the solid is a circular disk with radius r. Parallel cross-sections perpendicular to the base are squares.

This is a difficult problem from the text. If someone could walk me through how to do this problem using integration, I think it will help me grasp how to do the other problems similar to it.

I'm having trouble starting, and not sure what I'm supposed to do. I think I'm supposed to find the surface area of the squares with respect to their distance from the center of the disk, and evaluate it from the -r to r. But how to I find out what the area of the square is at any given distance from along the radius?
• Oct 13th 2009, 10:08 PM
MarkOfEternity
I think I've figured out how to start the problem. Please, someone, let me know if this is right.

I would look at the surface area of each slice, from 0 to r. Since each surface area is equal to the distance of how far that square is from r, we can call that point x, and integrate like this:
$\displaystyle \int{x^2} dr$, from 0 to r.
Is this right? If it is, how to I get x in terms of r?
• Oct 14th 2009, 05:08 AM
HallsofIvy
Quote:

Originally Posted by MarkOfEternity
I think I've figured out how to start the problem. Please, someone, let me know if this is right.

I would look at the surface area of each slice, from 0 to r. Since each surface area is equal to the distance of how far that square is from r, we can call that point x

x is a length, not a point. And 'area' is not equal to 'distance'. I am not sure what you are trying to say there.
Quote:

, and integrate like this:
$\displaystyle \int{x^2} dr$, from 0 to r.
Is this right? If it is, how to I get x in terms of r?
I think you may be misleading your self by calling the variables "r" and "x".
For one thing, it makes no sense to integrate with respect to r when r is given as a constant.

Draw the circular base with its center at (0,0) on an xy- graph. Its equation is $\displaystyle x^2+ y^2= r^2$. Now, look at the slice at each x. its area is $\displaystyle s^2$ where s is the length of an edge. And that is the difference of the two y-coordinates where the edge crosses the circle. What are those y coordinates in terms of x? What is s in terms of x?

The volume is [/tex]\int s^2 dx[/tex]. What are the limits of integration? What are the smallest and largest values of x in this circle?
• Oct 14th 2009, 06:43 AM
MarkOfEternity
Draw the circular base with its center at (0,0) on an xy- graph. Its equation is $\displaystyle x^2+ y^2= r^2$. Now, look at the slice at each x. its area is $\displaystyle s^2$ where s is the length of an edge. And that is the difference of the two y-coordinates where the edge crosses the circle. What are those y coordinates in terms of x? What is s in terms of x?
The volume is $\displaystyle \int s^2 dx$. What are the limits of integration? What are the smallest and largest values of x in this circle?
Okay, so $\displaystyle \int s^2 dx$ goes from -r to r, right?