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Volumes with Cross Sections

AP Calculus AB· difficulty 3/5

Base on [0,1][0, 1] has width f(x)=2xf(x) = 2x at each xx. Semicircular cross-sections perpendicular to xx-axis (with diameter the base width). Volume?

  • A

    2π3\tfrac{2\pi}{3}

  • B

    π3\tfrac{\pi}{3}

  • C

    π6\tfrac{\pi}{6}

    check_circle
  • D

    π\pi

Explanation

Radius r=f(x)/2=xr = f(x)/2 = x. Area = 12πx2\tfrac{1}{2}\pi x^2. V=∫01πx22 dx=π/6V = \int_0^1 \tfrac{\pi x^2}{2}\,dx = \pi/6.

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