Question

Multivariable Calculus

[A] Consider the region R in the first quadrant that is outside the circle r = 1 and inside the four-leaved rose r = 2 sin 2θ).

(A.1) Draw a sketch of the circle and the four-leaved rose (include the entire graph) and shade the region R. Feel free to use your graphing calculator.

(A.2) Write the following double integral as an iterated integral in polar coordinates. Do not evaluate the integral in this part. Be sure to use appropriate notation. (In order to find the interval for theta, you will have to find the TWO values of theta for which the circle and four-leaved rose intersect (in the first quadrant). Set the two functions equal to each other and solve the resulting equation; it should be a simple trig equation. Also note that the function that you are integrating, cos 2θ, is already written in polar form and thus will not need to be converted. Do not use decimal approximations for your angles; they should include a factor of π if you have found them correctly.)

∫_R ∫ cos 2θ dA

(A.3) Evaluate the integral in (A.2). Show all work!!! (After evaluating the inner integral, the outer integral should only require a U-substitution. Do not give a decimal approximation to the integral and do not use a computer program to calculate your antiderivatives and/or integrals.)

Answer 1