Problems

In Problems 1-16, find the volume using polar coordinates.

17            Find the volume of the solid over the cardioid r = 1 + cos θ between the plane z = 0 and the cone z = r.

18            Find the volume of the solid over the cardioid r = 1 + cos θ between the paraboloids z = r² and z = 8 - r².

19            Find the volume of the solid over the circle r = sin θ between the plane z = 0 and the hemisphere

20            Find the volume of the solid over the circle r = 2 cos θ between the plane z = 0 and the cone z - 1 - r.

21            Find the volume of the solid over the polar rectangle α ≤ θ ≤ β, a ≤ r ≤ b, between the plane z = 0 and the cone z = r.

22            Find the volume of the portion of the hemisphere over the polar rectangle α ≤ θ ≤ β, a ≤ r ≤ b (assuming b ≤ 1).

23            A circular object of radius b has density equal to the distance from the outside of the circle. Find (a) the mass, (b) the moment of inertia about the origin.

24            A circular object of radius b has density equal to the cube of the distance from the center. Find (a) the mass, (b) the moment of inertia about the origin.

25            Find the moment of inertia about the origin of a circular ring a ≤ r ≤ b, 0 ≤ θ ≤ 2n, of constant density k.

26            Find the moment of inertia of a circular object of radius b and constant density k about a point on its circumference. (The center can be put at (0, b), so the object is on the polar region 0 ≤ r ≤ 2b sin θ, 0 ≤ θ ≤ π.)

27            An object has constant density k on the circular sector 0≤x≤l,0≤y≤ Find (a) the center of mass, (b) the moment of inertia about the origin.

28            An object of constant density k covers the cardioid r ≤ 1 + cos θ, 0 ≤ θ ≤ 2π. Find

(a) the center of mass, (b) the moment of inertia about the origin.

29            An object of constant density k covers the region inside the circle r = 2b sin θ and outside the circle r = b. Find (a) the center of mass, (b) the moment of inertia about the origin.

30            An object of constant density k covers the polar region

0 ≤ θ ≤ π/2, 0≤r≤b sin 2θ.

Find (a) the center of mass, (b) the moment of inertia about the origin.

31

(a) Use polar coordinates to evaluate

(b)  Show that

(c)  Now evaluate the single integral