Name: ___________________

Math 122 Lab 1: Area in Polar Coordinates

Introduction Polar coordinates provide a different setting for calculus problems. For many problems, the solution of the problem is far easier in polar coordinates than in rectangular (x,y) coordinates. In this lab, you will review/learn polar coordinates and use them to solve an interesting area problem.

Problems Start by graphing the region bounded by , y = x and y = 2x. Also, graph the region bounded by , y = 2x and y = 4x. Sketch both regions here.

Based on your sketches, how do you think that these areas compare? You will compute the areas exactly later. For now, briefly explain why this would be a difficult calculation using the methods of section 5.1.

Polar coordinates A description of polar coordinates is in the book in section 0.7 starting on page 72. Instead of using the horizontal distance x and vertical distance y to describe a point, we use the distance r from the origin and angle θ from the positive x-axis. To sketch a picture showing the polar point (2,π/4), start by drawing the line with angle θ = π/4 and then move 2 units along that line. Show that here.

To convert back and forth between rectangular and polar coordinates, we use the equations x = r cos θ, y = r sin θ, and tan θ = y / x. Use the first two equations to find rectangular coordinates for the polar point (2,π/4) that you just plotted.

Next, show that the line y = x has polar equation tan θ = 1 or θ = π/4.

Show that y = 2x has polar equation .

Sketch the region in between the polar graphs θ = π/4 and .

Now, find a polar equation of the form r = f(θ) for the curve .

Verify that your equation is right by drawing the polar graph on your calculator. To do this, press MODE and change Graph to POLAR. Press Enter and then go to the Y= list. You should see prompts for r1 = and r2 = and so on. Enter the equation found above for f(θ) on the r1 line and graph it. Sketch your result here.

Before returning to the area problem, let’s have some fun with polar graphs. Sketch here.

Sketch here.

Sketch here.

Sketch for here. (A window of and is good.) Why is it called the “Garfield curve”?

Finally, sketch here.

In polar coordinates, the area between r = f(θ), θ = a and θ = b is given by . Use this formula and your calculator to find the area of the region bounded by , y = x and y = 2x.

Write down the integral and find the area of the region bounded by , y = 2x and y = 4x. How does it compare to the previous area?

For any positive number m, find the area of the region bounded by , y = mx and y = 2mx. Write down any integrals you have the calculator compute for you and in one or two sentences describe the result.