Math 152 Week 15 Workshop Problems 4/22/24 Write up your solutions to each of these problems on a seperate sheet of paper. 1. Consider the polar equation r = 1 - 2 sin(0), for 0≤0≤2π (a limacon). (a) Graph this equation as a Cartesian equation in the re- plane. (Your graph should look like a sine wave with some transformations done to it.) (b) Using your graph in part a) as a guide, sketch the graph of the above equation as a polar equation in the xy-plane. Then use a graphing utility to check your work. (c) Set up, but do not evaluate, a polar integral to calculate the area of the inner loop of the limacon. 2. Consider the two curves r₁ = 2 sin(0) and r2 = 2 sin(0) and r2 = 2√3 cos(0). r1 (a) Graph the two curves in the same xy-plane. (b) On your graph, shade the region in the first quadrant that is outside r₁ and inside r2. Then set up an integral to find its area. (c) On your graph, shade (with a different color or different pattern) the region in the first quadrant that is outside r2 and inside r₁. Then, set up an integral to find its area. 3. Consider the polar curve r = 2+2 cos(0). (a) Show that the Cartesian coordinate (0, 2) lies on this curve by converting the point into polar and showing that the polar coordinate satisfies the above equation. (b) Find the Cartesian equation of the line tangent to the polar curve at this point. (c) Find the arc length of this curve on the interval 0 ≤ 0≤πT.