368 CHAPTER 20. THE INTEGRAL IN OTHER COORDINATES

6. Suppose the density is 2+ x. Find the mass of the plate which is the inside of thepolar curve r = sin(2θ). Hint: This is one of those fussy things with negative radius.

7. The area density of a plate is given by λ = 1+ x and the plate occupies the inside ofthe cardioid r = 1+ cosθ . Find its mass.

8. The moment about the x axis of a plate with density λ occupying the region R isdefined as my =

∫R yλdA. The moment about the y axis of the same plate is mx =∫

R xλdA. If λ = 2− x, find the moments about the x and y axes of the plate insider = 2+ sin(θ).

9. Using the above problem, find the moments about the x and y axes of a plate havingdensity 1+ x for the plate which is the inside of the cardioid r = 1+ cosθ .

10. Use the same plate as the above but this time, let the density be (2+ x+ y). Find themoments.

11. Let D ={(x,y) : x2 + y2 ≤ 25

}. Find

∫D e25x2+25y2

dxdy. Hint: This is an integral ofthe form

∫D f (x,y)dA. Write in polar coordinates and it will be fairly easy.

12. Let D ={(x,y) : x2 + y2 ≤ 16

}. Find

∫D cos

(9x2 +9y2

)dxdy.Hint: This is an inte-

gral of the form∫

D f (x,y)dA. Write in polar coordinates and it will be fairly easy.

13. Derive a formula for area between two polar graphs using the increment of area ofpolar coordinates.

14. Use polar coordinates to evaluate the following integral. Here S is given in terms ofthe polar coordinates.

∫S sin

(2x2 +2y2

)dV where r ≤ 2 and 0≤ θ ≤ 3

2 π .

15. Find∫

S e2x2+2y2dV where S is given in terms of the polar coordinates r ≤ 2 and

0≤ θ ≤ π .

16. Find∫

Syx dV where S is described in polar coordinates as 1≤ r≤ 2 and 0≤ θ ≤ π/4.

17. Find∫

S

(( yx

)2+1)

dV where S is given in polar coordinates as 1 ≤ r ≤ 2 and 0 ≤θ ≤ 1

6 π .

18. A right circular cone has a base of radius 2 and a height equal to 2. Use polarcoordinates to find its volume.

19. Now suppose in the above problem, it is not really a cone but instead z = 2− 12 r2.

Find its volume.

20.3 Cylindrical And Spherical CoordinatesCylindrical coordinates are defined as follows.

x(r,θ ,z) ≡

 xyz

=

 r cos(θ)r sin(θ)

z

 ,

r ≥ 0,θ ∈ [0,2π),z ∈ R