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Math Help - surface area of rotating region

  1. #1
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    surface area of rotating region

    Find the area of the surface obtained by rotating the curve
    y=1+5x^2
    from x=0 to x=1 about the y-axis

    I know how to do it about the x-axis but how does the formula change if your rotating about the y axis

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  2. #2
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    Quote Originally Posted by dat1611 View Post
    Find the area of the surface obtained by rotating the curve
    y=1+5x^2
    from x=0 to x=1 about the y-axis

    I know how to do it about the x-axis but how does the formula change if your rotating about the y axis

    The formula you need is

    S=\int_{a}^{b}2 \pi x \sqrt{1+\left( \frac{dx}{dy}\right)^2}dy

    First to find the limits of integration we plug in x=0 and x=1 to get

    y=1 and y=6

    Now we solve the equaiton for x to get

    y=1+5x^2 \iff x=\sqrt{\frac{y-1}{5}}=\frac{1}{\sqrt{5}}(\sqrt{y-1})


    Now we take the derivative with respect to y to get

    \frac{dx}{dy}=\frac{1}{2\sqrt{5}}\frac{1}{\sqrt{y-1}}

    From here just plug into the formula and integrate.

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