Math 2374 Spring 2017 - Week 5
X Quiz 3 will cover 3.1, 5.1− 5.3.
Quick Reivew from previous lecture
Fact. (Fubini’s Theorem) If f is continuous on R = [a, b]× [c, d], then∫ ∫
R
f (x, y)dA =
∫ b
a
[∫ d
c
f (x, y)dy
]
dx =
∫ d
c
[∫ b
a
f (x, y)dx
]
dy.
1
Example 3. Evaluate the integral∫ ∫R
yexydA,
where R = [0, 1]× [0, 2].
7
Example 4. Evaluate the integral∫ ∫R
y3exy2
dA,
where R = [0, 1]× [0, 2].
8
5.3 Double Integralas over General Regions
In this section, we want to set up the double integral of f (x, y) over regions D:
2
y
x
y
x
(1) bounded by 2 functions of x
D:
(2) bounded by 2 functions of y
D:
Example 1. Evaluate the integral∫ ∫
R
(x3y + ex)dA,
where R is the triangle with vertices (0, 0), (π/2, 0), (π/2, π/2).
3
y
x
y
x
(y-simple) (x-simple)
Example 2. Integrate f (x, y) = xy2 over the region D which is bounded by
y = 3x and y = x2.
4
y
x
5.4 Changing the order of Integration
Sometimes, by changing the order of the iterated integral, we will have a much
easier integral to solve.
See EX2 in NOTE 5.1, 5.2.∫
1
0
∫
2
0
yexydydx
∫
2
0
∫
1
0
yexydxdy
5
