Multiple Integrals
Lesson · Advanced
Calculus
Area and Volume by Integrals
Choosing Limits of Integration
Choose the limits from inside to outside.
Find the volume of the shaded region.

Find the volume under
above the triangular region

The region is a pyramid with triangular base.
The limits are
and vertically,
Thus
After integrating with respect to z,
This gives
Find the volume of the region satisfying
The inequalities already tell us the order.
First, 0 ≤ x ≤ 2.
For a fixed x, 0 ≤ y ≤ x.
For fixed x, y, 0 ≤ z ≤ y.
Therefore
Integrating,
The cube 2 × 2 × 2 has volume
There are 3! orders for x, y, z, by symmetry.