Question

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`

Answer 1

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 `R` is defined as follows, it is specifically about the area within Q1 and Q2:

• `{(-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`