1. Section 16.1 Definite Integral of a Function of Two Variables

2. Recall: If f is continuous on an interval, [a,b], then f is integrable on [a,b] and

3. Now if we have f(x,y) continuous on some region then we have the following where ΔA = ΔxΔy This sums up the volume of mn rectangular solids This is the definition of the definite integral of f over R Let’s take a look at #4 on page 788

4. Setting up Riemann sums can be tricky Let’s try with problem 7 on page 788 then we will evaluate it with Maple

5. Interpretations of the Double Integral As we have seen, the double integral can be used to calculate the volume under a graph and above the region R

6. Double Integral as Area They can also be used to find areas –If we have the case where f(x,y) = 1 for all points (x,y) in the region R, each term of the Riemann Sum is 1·Δ A = ΔA and thus the double integral gives us the area of region R

7. Double Integral as Average Value As in the one variable case, we can use the definite integral to compute the average value of a function Find the average value of f if