Multivariate Calculus - Bruce E. Shapiro

LECTURE 18. DOUBLE INTEGRALS IN POLAR COORDINATES 149
Figure 18.3: The domain of integration of the integral in example 18.3.
We emphasize the fact that when the order of integration is reversed, the order of
the differentials dθdr reverses to drdθ and the order two integrals is reversed.
Finally, to solve the integral we arbitrarily choose one of these forms, say the
second one, to give
∫ √ 2
0
∫ √ 4−y 2
xdxdy =
y
=
∫ π/4
0
∫ π/4
0
= (8/3)
(
= (8/3)
cos θ
∫ 2
0
r 2 drdθ
( ∣ )
1 ∣∣∣
2
cos θ
3 r3 dθ
∫ π/4
0
0
cos θdθ
)
= (8/3)( √ 2/2) = 4 √ 2/3 ≈ 1.88562
sin θ| π/4
0
Example 18.4 Find the area of a circle of radius a using double integrals in polar
coordinates.
Solution. Consider a circle of radius a whose center is at the origin. Then the
interior of the circle is the set
C = {(r, θ) : 0 ≤ r ≤ a, 0 ≤ θ ≤ 2π}
Since the area of any region C is ∬ C
fdA, the area of the circle is
Math 250, Fall 2006 Revised December 6, 2006.