70.2K

Verified Solution

Link Copied!

Multivariable Calculus

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

(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 useyour graphing calculator.

(A.2) Write the following double integral as an iteratedintegral in polar coordinates. Do not evaluate the integral in thispart. Be sure to use appropriate notation. (In order to find theinterval for theta, you will have to find the TWO values of thetafor which the circle and four-leaved rose intersect (in the firstquadrant). Set the two functions equal to each other and solve theresulting equation; it should be a simple trig equation. Also notethat the function that you are integrating, cos 2?, is alreadywritten in polar form and thus will not need to be converted. Donot use decimal approximations for your angles; they should includea factor of ? if you have found them correctly.)

?_R ? cos 2? dA

(A.3) Evaluate the integral in (A.2). Show all work!!! (Afterevaluating the inner integral, the outer integral should onlyrequire a U-substitution. Do not give a decimal approximation tothe integral and do not use a computer program to calculate yourantiderivatives and/or integrals.)