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Math Help - vol

  1. #1
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    vol

    taking slices at fixed y, how do i find the vol of the solid enclosed by y=x^2, y+z=2 and z=1?
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  2. #2
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    You need a "floor" for your solid - I'm going to assume z=0.

    You divide the solid into two parts - the first part, with y going from 0 to 1, has the z=1 plane on the top. The second part, with y going from 1 to 2, has the y+z=2 plane on the top. All of the cross sections are rectangles.

    V_1=\int_0^1A\ dy=\int_0^1(1-0)(\sqrt{y}-(-\sqrt{y})\ dy=\int_0^12\sqrt{y}\ dy

    V_2=\int_1^2A\ dy=\int_1^2((2-y)-0)(\sqrt{y}-(-\sqrt{y})\ dy=\int_1^2(4-2y)\sqrt{y}\ dy

    where I substituted length times width for the area in both cases. Of course, the total volume is V=V_1+V_2.

    Post again in this thread if you're still having trouble.

    - Hollywood
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  3. #3
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    may i know how ((2-y)-0) for V2 is obtained? i thought that the lenght for y would be (2-1)?

    Also, how do you know that we can use a single integral to find the vol? i thought that double integrals are mean for areas and triple integrals are meant for vol?

    thanks!
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  4. #4
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    Quote Originally Posted by alexandrabel90 View Post
    may i know how ((2-y)-0) for V2 is obtained? i thought that the lenght for y would be (2-1)?

    Also, how do you know that we can use a single integral to find the vol? i thought that double integrals are mean for areas and triple integrals are meant for vol?

    thanks!
    For V_1, the "roof" is the plane z=1, but for V_2, the "roof" is the plane y+z=2, which intersects the plane z=1 at y=1 and slopes down to intersect the "floor" z=0 at y=2. The height for any given y between 1 and 2 is given by z=2-y.

    When I substituted length times width for the cross-sectional area, I actually did an (extremely easy) double integral. So that makes three integrals for a volume.

    - Hollywood
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  5. #5
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    does that mean that i can actually find volume of a surface via single, double or triple integrals?
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  6. #6
    Senior Member AllanCuz's Avatar
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    Quote Originally Posted by alexandrabel90 View Post
    does that mean that i can actually find volume of a surface via single, double or triple integrals?
    Yes. A single integral is a simplified double integral. A double intregal is a simplified triple integral. If you can see it right away, there is no need to use any more then 1 or 2.
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