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Math Help - Find the volume of the described solid.

  1. #1
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    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?
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  2. #2
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    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:
    \int{x^2} dr, from 0 to r.
    Is this right? If it is, how to I get x in terms of r?
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  3. #3
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    Quote Originally Posted by MarkOfEternity View Post
    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.
    , and integrate like this:
    \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 x^2+ y^2= r^2. Now, look at the slice at each x. its area is 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?
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  4. #4
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    Quote Originally Posted by HallsofIvy View Post
    Draw the circular base with its center at (0,0) on an xy- graph. Its equation is x^2+ y^2= r^2. Now, look at the slice at each x. its area is 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 \int s^2 dx. What are the limits of integration? What are the smallest and largest values of x in this circle?
    Okay, so \int s^2 dx goes from -r to r, right?

    But then... at 0, s=r, and at r and -r, s=0. How do I put s in terms of x?
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