Posted by Sophie on Thursday, January 21, 2010 at 3:14am.
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  Reiny, Thursday, January 21, 2010 at 9:24am
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 xaxis, and the yaxis 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 (pi2)/4 = .2854
1/3 = .33333
So who is bigger?
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