# HW #15 - Double Integrals ```Double Integrals
Recommended Problems: 5.1: 5, 7, 8, 11, 13;
5.2: 1, 3, 5, 7, 9, 10, 11, 13, 22, 26, 27, 28, 29
Evaluate each of the following if R  [1,1]  [0, 2] :
1)
 ( x
2)
 (ax  by  c) dA (a, b, c arbitrary constants)
R
2
 y3 ) dA
R
3) Find the volume of the solid whose top is z  xy cos( x 2 ) , whose base is the rectangle
[0,1]  [0, 2] and whose sides are vertical.
4) Let R be the region bounded by the y-axis and the parabola x = 4y - y2.
Find the integral over R of f(x, y) = xy.
5) A certain double integral over a region R is computed as
 /6
cos x
0
sin x
 
sec x dy dx . Plot the
region R and calculate the value of the integral.
6) Show that if f is defined on [a, b] and g is defined on [c, d], then over the rectangle R =
[a, b]  [c, d],
7) Evaluate

R
 e
b
d
a
c
f ( x) g ( y) dA  ( f ( x) dx)( g ( y) dy) .
x y
R
dA where R is the interior of the triangle whose vertices are (0, 0),
(1, 3) and (2, 2).
8) Write the integral obtained when you change the order of integration in
1
 
1
1 | y|
f ( x, y ) dx dy .
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