Math 21a Homework 05 Solutions Spring, 2014
1. In each part, DRAW the region D, and evaluate the integral.
(a) (Stewart 12.3 #8 )ZZD
y
x5+ 1 dA, where Dis the region D={(x, y)|0≤x≤1,0≤y≤x2}.
Solution:
Here is a sketch of D:
1
x
1
y
We integrate as follows:
ZZD
y
x5+ 1 dA =Z1
0Zx2
0
y
x5+ 1 dy dx
=Z1
0
1
x5+ 1 ·y2
2
y=x2
y=0
dx
=1
2Z1
0
x4
x5+ 1 dx =1
2·1
5ln |x5+ 1|
1
0
=1
10 (ln 2 −ln 1) = 1
10 ln 2
(b) (Stewart 12.3 #10 )ZZD
x3dA, where Dis the region D={(x, y)|1≤x≤e , 0≤y≤ln x}.
Solution:
Here is a sketch of D:
y
We integrate as follows:
ZZD
x3dA =Ze
x3dy dx
