Double integration in polar coordinates problems and solutions
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Double integration in polar coordinates problems and solutions
Double integration in polar coordinates. 1. 1. Compute. R f(x, y) dx dy, where f(x,
y) = x2 + y2 and R is the region inside the circle of radius 1, centered at (1,0).
Double integration in polar coordinates �� 1.
Compute R
1
f (x, y) dx dy, where f (x, y) = �
x2
+ y2
and R is the region inside the
circle of radius 1, centered at (1,0). Answer: First we sketch the region R y r = 2 cos θ x
1
Both the integrand and the region support using polar coordinates. The equation of the circle in polar coordinates is r = 2 cos θ, so using radial stripes the limits are (inner) r from 0 to 2 cos θ; (outer) θ from −π/2 to π/2. Thus, �� � π/2 � 2 cos θ � π/2 � 2 cos θ 1 f (x, y) dx dy = r dr dθ = dr dθ. r R −π/2 0 −π/2 0 Inner integral: 2 cos θ.
y
π/2
Outer integral: 2 sin θ|−π/2 = 4. 2. Find the area inside the cardioid r = 1 + cos θ.
1
r = 1 + cos θ
Answer: The cardioid is so-named because it is heart-shaped. Using radial stripes, the limits of integration are (inner) r from 0 to 1 + cos θ; (outer) θ from 0 to 2π. So, the area is �� � 2π � 1+cos θ dA = r dr dθ. R
0
0
(1 + cos θ)2 Inner integral: . 2 Side work: � � � 2π 1 + cos 2θ θ sin 2θ 2 cos θ dθ = dθ = + +C ⇒ cos2 θ dθ = π. 2 2 4 0 Outer integral: � 2π 0
(1 + cos θ)2 = 2
The area of the cardioid is
3π . 2
2π
� 0
1 cos2 θ π 3π + cos θ + dθ = π + 0 + = . 2 2 2 2
2
x
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18.02SC Multivariable Calculus Fall 2010
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