How to solve a double integration

Hello everyone, welcome to my first post. Today I'm going to solve a double integration.

Find the volume of the solid bounded above by the plane z = 4 − x − y and below

by the rectangle R = {(x, y) : 0 ≤ x ≤ 1, 0 ≤ y ≤ 2}.

Solution

The volume under any surface z = f(x, y) and above a region R is given by

f(x,y) is, in our case, 4-x-y. And the limits on the integration are very simple: in the variable x, the limits are 0 and 1, and the y is from 0 to 2. We resolution is simple: You solve one of the integrals(in this problem, we'll solve the x first), considering the other variable a constant. With the first integral done, just solve the area left with the limits.

Repeat to solve the integral of y and the solution is done.

This double integral  are the easiest types to evaluate because all four limits of integration are constants. This happens when the region of integration is rectangular in shape. In non-rectangular regions of integration the limits are not all constant so we have to get used to dealing with non-constant limits.

The 3d plot has been done by Wolfram Alpha. It's a good site to see the solutions if you're stuck doing a math problem.