double integral

Rectangular Domain

Suppose we want to compute the volume under a plane defined by the multivariate function , in the rectangular region ( and )

We can switch the order of and

Compute the volume under the slanted plane in the region ,

Step 1: Visualize

Step 2: Write the double integral for volume

Step 3: Compute the inner integral

Consider constant:

x^{2}+yx\bigg|_{0}^{2} = 2+2y$$

Step 4: Compute the outer integral

Plug step 3 back in

Summary: Quick solution

Intuition

Non-rectangular domain

Suppose we want to integrate the function for the domain and .

Approach 1: Relate y in terms of x

We want to rewrite where

or in general,

Then

Triangular domain

Approach 2: Relate x in terms of y

Relate x in terms of y: Then

Approach 3: Geometry

This depends on the shape of the domain