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Area Between Curves

AP Calculus AB· difficulty 3/5

Area between y=sin⁡xy = \sin x and y=cos⁡xy = \cos x on [0,π/4][0, \pi/4] is

  • A

    2−1\sqrt 2 - 1

    check_circle
  • B

    2\sqrt 2

  • C

    1−21 - \sqrt 2

  • D

    11

Explanation

On this interval cos⁡x≥sin⁡x\cos x \ge \sin x. ∫0π/4(cos⁡x−sin⁡x) dx=[sin⁡x+cos⁡x]0π/4=2/2+2/2−0−1=2−1\int_0^{\pi/4}(\cos x - \sin x)\,dx = [\sin x + \cos x]_0^{\pi/4} = \sqrt 2/2 + \sqrt 2/2 - 0 - 1 = \sqrt 2 - 1.

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