posted by Sophie on .
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.
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
√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
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?