# How to Draw the R region of integration and change the order of integration (do NOT evaluate the integral)?

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#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#