MATH 20C Lecture 25

15.1 Double Integral

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Recall

Double Integral the volume under the graph of over

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Now let’s define the double integral using Riemann Sum

Step 1 (subdivisions):

• Subdivide into subintervals of the same length
• Subdivide into subintervals of the same length
  This creates an grid of subrectangles in .

Step 2:
Choose a sample point in each

Step 3:
Volume under the graph over

Define the Riemann Sum

This approximates the volume under the graph of over

Double Integral of $f(x,y)$

The double integral of over a rectangle is defined by
If the limit exists, we say is integrable over

Fact

If is continuous on , then is integrable on .

Example. $f(x,y)$ $R=[1,4]\times [1,3]$ $N=3,M=2$

Riemann sum

Theorem

Assume that and are continuous over a rectangle . Then

1. 
   
2. For any constant
   
3. 
   

How to compute double integrals?

Iterated integrals

View as a constant for the inner integral and integrate
is a function of

View as a constant for the inner integral and integrate
is a function in

Example. ${\int }_{x=0}^1{\int }_{y=0}^{\frac{\pi }{6}}{x}^3\cos ydydx$

Example. ${\int }_0^1{\int }_0^2(x^2-\sin y)dxdy$

Fubini's Theorem

Let be a continuous function on

Example. Evaluate $\int \int x{e}^{xy}dA$ where $R=[0,1]\times [0,1]$

Second double integral is easier