Change the order of a double integration

Question

How should I change the order of the double integral of : $\int _0^1\:\int _{-x}^x 1\,dy\,dx$

Answer 1

Look at the picture here:

Since in the first integral you fixed $x$ and let $y$ vary from $-x$ to $x$ then the region of integration must be the triangle between vertical line $x = 1$ and the lines $y = x$ (blue one) and $y = -x$ (the red one).

Here a picture that shows how you integrate in the first integral:

Then you can now fix $y$ and you must see that $x$ vary from the absolute value of y to 1.

Here the picture:

Answer 2

Draw a picture ! Then you should see:

$$\int _0^1\:\int _{-x}^x\:1\:dydx=\int_{-1}^1\:\int _{|y|}^1\:1\:dxdy.$$

Answer 3

The integration range is defined by$$0\le x\le1,\,-x\le y\le x,$$or equivalently$$-1\le y\le1,\,|y|\le x\le1,$$so the double integral is $$\int_{-1}^1\int_{|y|}^1dxdy=\int_{-1}^1(1-|y|)dy=2\int_0^1(1-y)dy=1,$$in agreement with the original $\int_0^12xdx=1$.