MATH 20C Lecture 26

15.2 - Double Integral Over More General Regions

Theorem

Let be a continuous on . Suppose is a product. Then

Proof:

Example. $\int \underset{R}{\int }\frac{x}{y}dA$ over $R=[-1,2]\times [1,2]$

$D$ Domain

= volume under the graph of and over
One can define it precisely using Riemann sum.

Theorem

Assume that and are integrable on a domain . Then

1. 
   
2. For any constant
   

Regions between two graphs

• Each vertical strip is labeled by its x-coord
• The range of the vertical strips is from to
• For a vertical strip at the bounds for are from to
  

Example. Evaluate $\int \underset{D}{\int }{e}^{x+y}dA$ over $D$ defined by $x\ge 1,y\ge 1,x+y\le 5$

Finish the computation next time…