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Math Help - Application of integration: Volumes

  1. #1
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    Application of integration: Volumes

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    im supposed to find the volume generated by rotating the given region about the specified line

    a) R1 about OC
    b) R1 about BC
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  2. #2
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    Hello, sobadin!

    Find the volume generated by rotating the region between y = x^3\text{ and }y = x^{\frac{1}{2}}

    (a) about the y-axis

    Washers: . V \;=\;\pi\int^1_0\left(y^{\frac{1}{3}} - y^2\right)\,dy

    Shells: . V \;=\;2\pi\int^1_0x\left(x^{\frac{1}{2}} - x^3\right)\,dx




    (b) about  y = 1

    Washers: . V \;=\;\pi\int^1_0\bigg[\left(1 - x^3\right)^2  - \left(1 - x^{\frac{1}{2}}\right)^2\bigg]\,dx

    Shells: . V \;=\;2\pi\int^1_0(1-y)\left(y^{\frac{1}{3}} - y^2\right)\,dy

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