Changing integration order and Fubini

When to choose this method

• The inner integral is difficult, but the region can be described by slices in the other direction.
• Check continuity or integrability before interchanging. Split the region when slice boundaries change.

Before calculating

1. Convert bounds into inequalities and sketch the boundaries.
2. Project onto the new outer axis and determine the inner range.

A complete example, step by step

1. The inner differential is dy: x is fixed while y runs from x to 1.

   $$D=\{(x,y):0\le x\le1,\ x\le y\le1\}$$

2. The region lies above y=x and below y=1 in the first quadrant.

   $$0\le x\le y\le1$$

3. For a horizontal slice, y runs from 0 to 1 and x runs from 0 to y.

   $$I=\int_0^1\int_0^y e^{y^2}\,dx\,dy$$

4. The integrand does not depend on x. The inner integral is the function value times the width y.

   $$\int_0^y e^{y^2}\,dx=\left[xe^{y^2}\right]_0^y=ye^{y^2}$$

5. The exponent derivative is now present: use u=y². The transformed bounds are still 0 and 1.

   $$I=\frac12\int_0^1 e^u\,du=\frac{e-1}{2}$$

Now try it yourself

Solve on paper before opening the hint or answer. These are self-assessment activities; to check a typed answer, open Learn.

Connect with other methods