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Thread: Shell Method Help

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    Shell Method Help

    I've been stuck on these problems and was hoping to have someone solve them so I could see how each one is done. Consider the curves given by y=sinx and cosx
    For each of the following problems, you should include a sketch of the region/solid being considered, as well as a labeled representative slice.
    1. Set up a definite integral whose value is the exact volume of the solid of revolution generated by revolving about the y-axis
    2. Set up a definite integral whose value is the exact volume of the solid of revolution generated by revolving about y=2

    Consider the finite region y = 1+ 0.5(x-2)^2 and y = 0.5x^2, and x=0
    3. Deterimine an expression involving one or more whose value of the volume of revolution generated by revolving the region R about the y-axis.

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    Re: Shell Method Help

    Quote Originally Posted by donutstaste View Post
    I've been stuck on these problems and was hoping to have someone solve them so I could see how each one is done. Consider the curves given by y=sinx and cosx
    For each of the following problems, you should include a sketch of the region/solid being considered, as well as a labeled representative slice.
    1. Set up a definite integral whose value is the exact volume of the solid of revolution generated by revolving about the y-axis
    2. Set up a definite integral whose value is the exact volume of the solid of revolution generated by revolving about y=2


    For #1 and #2, define the x-interval of each region to be rotated ...



    Consider the finite region y = 1+ 0.5(x-2)^2 and y = 0.5x^2, and x=0
    3. Deterimine an expression involving one or more whose value of the volume of revolution generated by revolving the region R about the y-axis.
    $1+0.5(x-2)^2 = 0.5x^2 \implies x = 1.5$

    $\displaystyle V = 2\pi \int_0^{1.5} x[1+0.5(x-2)^2 - 0.5x^2] \, dx$
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