1. Double Integrals
Introduction

2. Volume and Double Integral
z=f(x,y) ≥ 0 on rectangle R=[a,b]×[c,d]
S={(x,y,z) in R3 | 0 ≤ z ≤ f(x,y), (x,y) in R}
Volume of S = ?

4. Sample point (xij*, yij*)
ij’s column:
f (xij*, yij*)
Δ y
Δ x x y z
Rij
(xi, yj)
Sample point (xij*, yij*) x y
Area of Rij is Δ A = Δ x Δ y
Volume of ij’s column:
Total volume of all columns:

5. Definition

6. Definition: The double integral of f over the rectangle R is
if the limit exists
Double Riemann sum:

7. Note 1. If f is continuous then the limit exists and the integral is defined
Note 2. The definition of double integral does not depend on the choice of sample points
If the sample points are upper right-hand corners then

8. Example 1
z=16-x2-2y2
0≤x≤2
0≤y≤2
Estimate the volume of the solid above the square and below the graph

9. m=n=16
m=n=4
m=n=8 V≈
V≈41.5
V≈44.875
Exact volume?
V=48

10. Example 2 z

11. Integrals over arbitrary regions
A is a bounded plane region
f (x,y) is defined on A
Find a rectangle R containing A
Define new function on R: A
f (x,y) R

12. Properties Linearity Comparison
If f(x,y)≥g(x,y) for all (x,y) in R, then

13. Additivity A1 A2
If A1 and A2 are non-overlapping regions then
Area

14. Computation
If f (x,y) is continuous on rectangle R=[a,b]×[c,d] then double integral is equal to iterated integral a b x y c d y
fixed
fixed x

15. More general case
If f (x,y) is continuous on A={(x,y) | x in [a,b] and h (x) ≤ y ≤ g (x)} then double integral is equal to iterated integral y
g(x) A
h(x) x a x b

16. Similarly
If f (x,y) is continuous on A={(x,y) | y in [c,d] and h (y) ≤ x ≤ g (y)} then double integral is equal to iterated integral y d A y
h(y)
g(y) c x

17. Note
If f (x, y) = φ (x) ψ(y) then

18. Examples
where A is a triangle with vertices (0,0), (1,0) and (1,1)