20.2. EXERCISES 367

Example 20.1.4 Let R denote the inside of the cardioid r = 1+ cosθ for θ ∈ [0,2π]. Find∫R

xdA

Here the convenient increment of area is rdrdθ and so the integral is∫ 2π

0

∫ 1+cos(θ)

0xrdrdθ

Now you need to change x to the right coordinates. Thus the integral equals∫ 2π

0

∫ 1+cos(θ)

0(r cos(θ))rdrdθ =

54

π

A case where this sort of problem occurs is when you find the mass of a plate given thedensity.

Definition 20.1.5 Suppose a material occupies a region of the plane R. The density λ is anonnegative function of position with the property that if B⊆ R, then the mass of B is givenby∫

B λdA. In particular, this is true of B = R.

Example 20.1.6 Let R denote the inside of the polar curve r = 2+ sinθ . Let λ = 3+ x.Find the total mass of R.

As above, this is ∫ 2π

0

∫ 2+sin(θ)

0(3+ r cos(θ))rdrdθ =

272

π

20.2 Exercises1. Sketch a graph in polar coordinates of r = 2+sin(θ) and find the area of the enclosed

region.

2. Sketch a graph in polar coordinates of r = sin(4θ) and find the area of the regionenclosed. Hint: In this case, you need to worry and fuss about r < 0.

3. Suppose the density is λ (x,y) = 2− x and the region is the interior of the cardioidr = 1+ cosθ . Find the total mass.

4. Suppose the density is λ = 4− x− y and find the mass of the plate which is betweenthe concentric circles r = 1 and r = 2.

5. Suppose the density is λ = 4− x− y and find the mass of the plate which is insidethe polar graph of r = 1+ sin(θ).