SOLUTION: the area by the curve y^2 = x(2-x) is given by?

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Question 1029007: the area by the curve y^2 = x(2-x) is given by?
Answer by robertb(5830) About Me  (Show Source):
You can put this solution on YOUR website!
y%5E2+=+x%282-x%29 has domain the closed interval [0,2]. (Outside it, y^2 = x(2-x)is negative, which is impossible. Verify!!)
==>Area =
=.
Now let x+-+1+=+sin%28theta%29.
If x = 2, sin%28theta%29+=+1,==> theta+=+pi%2F2
If x = 0, sin%28theta%29+=+-1,==> theta+=+-pi%2F2.
Also, d%28x-1%29+=+dx+=+cos%28theta%29d%28theta%29
==> Area = 2%2Aint%28cos%28theta%29%2A+cos%28theta%29%2Cd%28theta%29%2C+-pi%2F2%2Cpi%2F2%29
=2%2Aint%28cos%5E2%28theta%29%2Cd%28theta%29%2C+-pi%2F2%2Cpi%2F2%29
=2%2A%281%2F2%29int%281%2B+cos%282theta%29%2Cd%28theta%29%2C+-pi%2F2%2Cpi%2F2%29
=
= pi%2F2%2B%281%2F2%29sin%28pi%29-%28-pi%2F2+%2Bsin%28-pi%29%29
= pi%2F2+%2Bpi%2F2+=+pi sq. units.