Problems: Double Integration in Polar Coordinates | Answer Key - Edubirdie

University:
Massachusetts Institute of Technology
Course:
18.02SC | Multivariable Calculus
Academic year:

2022
Views:

22

Pages:

1
Author:

ohkawaiinn

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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