Thomas Calculus 13th [Solutions]

Section 15.4 Double Integrals in Polar Form 1109
44. The area in polar coordinates is given by
where r f ( )
f ( ) 2
f ( )
1 2 1 2
A r dr d r d f d r d
0 2
0
2 2
( ) ,
1 2 a
2 2 2 1
2 a 3 2 2
a 0 0 a 0 0
4 3
2 2 2 2 2 2 2
1
2
a 2a h cos a h 1
2 2ah cos 1 2ah
sin
d
a h d a h
2
a 0 4 3 2 0 4 3 2 4 3 2
0
45. average ( r cos h) r sin r dr d r 2r h cos rh dr d
2 2
1
2
2 2
a h
2
46.
A
3 /4 2sin
/4 csc
r dr d
1
3 /4 2 2
4sin csc d
2 /4
1
3 /4
2 sin 2 cot
2 /4 2
47. The region R is shaded in the graph below.
y
2
1
2 1 0 1 2
x
The polar equation of the outer circle is just r 2. The inner circle is
2
is equivalent to r 2rsin or r 2sin . The integrand is r in polar coordinates, so
2 2 2 2
x ( y 1) 1 or x y 2 y . This
R
2 2
x y dA r r dr d
0
2
2sin
3
2
r
8
d 1 sin
3 3
0 2sin
0
3
d
Write the integrand as
8 1 sin 1 cos
2 . The indefinite integral is then
3
8 1 3 8
definite integral is cos cos (3 4)
3 3 9
0
8 1 3
cos cos
3 3
and the
Copyright
2014 Pearson Education, Inc.