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Volumes with Cross Sections

AP Calculus AB· difficulty 3/5

Base bounded by y=1−x2y = \sqrt{1 - x^2} and xx-axis (semicircle). Cross-sections perpendicular to xx-axis are semicircles with diameter on the base. Volume?

  • A

    π6\tfrac{\pi}{6}

    check_circle
  • B

    2π3\dfrac{2\pi}{3}

  • C

    π3\tfrac{\pi}{3}

  • D

    23\tfrac{2}{3}

Explanation

Diameter =1−x2= \sqrt{1-x^2}, radius =1−x2/2= \sqrt{1-x^2}/2. Area =12πr2=π8(1−x2)= \tfrac{1}{2}\pi r^2 = \tfrac{\pi}{8}(1-x^2). V=∫−11π8(1−x2) dx=(π/8)(4/3)=π/6V = \int_{-1}^{1}\tfrac{\pi}{8}(1-x^2)\,dx = (\pi/8)(4/3) = \pi/6.

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