Calculus

Find the area of the region bounded by the curves y = sin x, y = csc^2x, x = pi/4, and x = (3pi)/4.

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  1. A quick peek at our problem:
    https://www.wolframalpha.com/input/?i=plot+y+%3D+sin+x%2C+y+%3D+csc%5E2x

    looks like they intersect at x = π/2 and there is symmetry
    so we can take
    area = 2∫ (csc^2 x - sinx) dx from π/4 to π/2
    recalling that the derivative of cotx = -sec^2 x

    = 2[ (- cotx + cosx) ] from π/4 to π/2
    = 2 ( (-cot π/2 + cos π/2 - (-cot π/4 + cos π/4) )
    = 2( 0 + 0 - (-1 + √2/2) )
    = 2(1 - √2/2)
    = 2 - √2

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    Reiny
  2. Thank you!

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