How to Draw the R region of integration and change the order of integration (do NOT evaluate the integral)?
#int_0^1int_-sqrt(1-y^2)^sqrt(1-y^2)dxdy#
1 Answer
Jul 30, 2018
This integration is all about the unit circle, because the underlying relationship is:
#x^2 + y^2 = 1 implies {(x = +-sqrt(1 - y^2)),(y = +-sqrt(1 - x^2)):}#
Here, because
#{(-sqrt(1-y^2) le x le sqrt(1-y^2)),(qquad qquad qquad 0 le y le 1):}#
The order can be reversed as:
#{(0 le y le sqrt(1-x^2)),(-1 le x le 1):}#
So:
#int_0^1 int_-sqrt(1-y^2)^sqrt(1-y^2) \ dx \ dy = int_-1^1 int_0^sqrt(1-x^2) \ dy \ dx#