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March 24, 2017

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Which has more area, the region in the first quadrant enclosed by the line x+y=1 and the circle x^2+y^2=1, or the region in the first quadrant enclosed by the line x+y=1 and the curve sqrt(x)+sqrt(y)=1? Justify your answer.

  • calculus - ,

    Nice Question!!!

    The first one we can do without Calculus, it is simply the segment between the chord from (0,1) to (1,0), that is
    pi/4 - 1/2 = (pi - 2)/4

    the second equation is
    √x + √y = 1
    or
    √y = 1-√x
    y = (1-√x)^2 = 1 - 2√x + x for 0 <= x <= 1

    the integral of 1 - 2√x + x is x - (4/3)x^(3/2) + (1/2)x^2

    and the area enclosed by √x + √y = 1 , the x-axis, and the y-axis is (from 0 to 1)
    (1 - 4/3 + 1/2) - 0
    = 1/6

    So the area between the curve √x + √y = 1 and the line x+y=1 is
    1/2 - 1/6 = 1/3

    then (pi-2)/4 = .2854
    1/3 = .33333

    So who is bigger?

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