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 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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