1. e2
log(x)
Example Consider the integral
x(e2y – 2e4y)10 dy dx = 1
Observe the difficulty with evaluating the integral using the given order of integration. Reverse the order of integration, and evaluate the integral. y
(e2 , 2)
y = log(x) 2 e2
x(e2y – 2e4y)10 dx dy = x
(1,0)
(e2,0) ey e2 2 2
x2(e2y – 2e4y)10
————— dy = 2
(e4 – e2y)(e2y – 2e4y)10
————————— dy = 2
x = ey

2. 2 2
(e4 – e2y)(e2y – 2e4y)10
————————— dy = 2
– (e2y – 2e4y)11
—————— = 44
y = 0
1 – (e4 – 4e4)11
—————— = 44
e44
———— . 44

3. Look at the Mean Value Theorem for Double Integrals on page 352 of the textbook, and use the theorem to find bounds on the integral 1
———— dx dy where D is the unit circle of radius 1/8
1 + x2 + y2
centered at the origin. D 1
Since  ————  , then
1 + x2 + y2 64 — 65 1 1
———— dx dy
1 + x2 + y2  — 65  — 64   D