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Math Help - volume about the y axis

  1. #1
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    volume about the y axis

    Hi, the questions is to find the volume of the solid formed when the curve y=x^3 rotated about the y axis between y=0 and y=1. I am having trouble integrating once x^2 has been made the subject of the equation.
    y=x^3
    y/x=x^3/x
    therefore
    x^2=y/x

    Is this right? If yes, how do you integrate an expression with both an x and y value? I haven't come across this before. Cheers
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  2. #2
    o_O
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    We'll use the disc method. Since we're rotating around the y-axis, we're going to be integrating with respect to y

    So, expressing the equation in terms of y: y = x^3 \ \Leftrightarrow \ x = \sqrt[3]{y}

    Each disk has a radius of x. So the area of each disk in terms of y, i.e. A(y), is \pi x^2 = \pi \left(\sqrt[3]{y}\right)^2 = \pi y^{\frac{2}{3}}

    So its volume is given by: V = \int_{y = 0}^{y = 1} A(y) dy = \int_0^1 \pi y^{\frac{2}{3}} dy
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