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Volumes by Disk and Washer Methods

AP Calculus AB· difficulty 3/5

Region between y=xy = x and y=x2y = x^2 rotated about y=0y = 0:

  • A

    2π3\tfrac{2\pi}{3}

  • B

    2π15\tfrac{2\pi}{15}

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  • C

    π6\tfrac{\pi}{6}

  • D

    π15\tfrac{\pi}{15}

Explanation

On [0,1][0, 1], y=xy = x is outer, y=x2y = x^2 is inner. V=π∫01(x2−x4) dx=π(1/3−1/5)=2π/15V = \pi\int_0^1 (x^2 - x^4)\,dx = \pi(1/3 - 1/5) = 2\pi/15.

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