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Math Help - Volumes of Revolution

  1. #1
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    Volumes of Revolution

    Consider the region bounded by y = x2 + 7, x = 3, the x-axis and the y-axis.

    Rotate the region about the y-axis and show how to find the volume of the resulting solid.

    Any help? (If anyone could point me to a good tutorial, that would be cool too)

    Peter
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  2. #2
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    Have you drawn the graph? It's pretty straight forward, it's a basic disk method problem. Read this:

    Pauls Online Notes : Calculus I - Volumes of Solids of Revolution / Method of Rings

    \pi\int_0^3(x^2+7)^2dx
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  3. #3
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    Quote Originally Posted by flybynight View Post
    Consider the region bounded by y = x2 + 7, x = 3, the x-axis and the y-axis.

    Rotate the region about the y-axis and show how to find the volume of the resulting solid.

    Any help? (If anyone could point me to a good tutorial, that would be cool too)

    Peter
    method 1 ... cylindrical shells w/r to x

    \displaystyle V = 2\pi \int_0^3 x(x^2+7) \, dx

    method 2 ... lower cylinder (y = 0 to y = 7) + upper washers (y = 7 to y = 16) w/r to y

    \displaystyle V = 63\pi + \pi \int_7^{16} 3^2 - (y-7) \, dy

    method 3 ... large cylinder (from y = 0 to y = 16) - paraboloid from disks (from y = 7 to y = 16)

    \displaystyle 144\pi - \pi \int_7^{16} y - 7 \, dy
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  4. #4
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    Thanks skeeter. Method 3 is what the prof has.
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