University:
University of Michigan
Course:
MATH 116 | Calculus II
Academic year:

2020
Views:

167

Pages:

1
Author:

D Robbins

Math 116 / Exam 2 (November 14 , 2012) page 7 5. [12 points] Solve each of the following problems. a. [4 points] Give inequalities for r and θ that describe the region shown below in polar coordinates. The region is bounded by the circle x2 + y 2 = 1, the line y = x, the x-axis and the vertical line x = 1.5. Solution: The line y = x is the polar line θ = π4 , so the limits for θ are 0 ≤ θ ≤ π4 . The 1.5 values of r range from r = 1 on the circle to the line r cos θ = 1.5, or r = cos θ . So the region is ( 0 ≤ θ ≤ π4 1.5 1 ≤ r ≤ cos θ b. [8 points] The functions in polar coordinates r = 2 sin θ and r = 4 cos θ represent the circles shown below Let A be the area of the intersection of these circles. Find an expression involving definite integrals in polar coordinates that computes the value of A. You do not need to evaluate the integrals. Solution: The curves intersect where 2 sin θ = 4 cos θ, or tan θ = 2. On the interval 0 ≤ θ ≤ arctan(2), r = 2 sin θ is the outside curve. On the interval arctan(2) ≤ θ ≤ π2 , r = 4 cos θ is the outside curve. The area is Z Z π 1 arctan(2) 1 2 2 A= 16 cos2 θdθ. 4 sin θdθ + 2 0 2 arctan(2)