MATH 2433 (Section 20708)

TA: Jeric Alcala

Office: PGH 612

Email: jsalcala@uh.edu

1. Set up a double integral in polar coordinates that gives the volume of the solid described

in rectangular coordinate system.

(a) solid bounded below by the xy−plane, above by the plane z = 2− x and on the sides

by the cylinder x2 + y2 = 4

(b) solid bounded by paraboloid z = 4− x2 − y2, cylinder x2 + y2 = 1 and plane z = 0

(c) solid on the first octant bounded by sphere x2 + y2 + z2 = 9

2

LabPop11a # 1:

Find the volume of the solid bounded above by z = 9−x2−y2 over the region x2 +y2 = 1.

a. 11π/2

b. 13π/2

c. 15π/2

d. 17π/2

e. None of these.

3

2. Find the center of mass of each given density λ:

(a) bounded by y = x2 and y = 4; λ(x, y) = 1

(b) bounded by y = x2 and y = x3; λ(x, y) = xy

(c) bounded by xy = 2, x−axis, x = 1 and x = 4; λ = 2x

4

LabPop11a # 2:

Find the x−coordinate of the center of mass of a plate of constant density bounded by

y = 2x, x = 1 and x−axis.

a. 1/3

b. 2/3

c. 2/5

d. 3/5

e. None of these.

5

3. Evaluate the triple integral of F (x, y, z) = x + 2z over the tetrahedron bounded by z =

y − x, z = 0 and y = 1

6

4. Set up (and evaluate) a triple integral that gives the volume of the solid described:

(a) solid enclosed by the surfaces z = x2 + y2 and z = 8− x2 − y2

(b) solid bounded by the plane y+z = 1 above the region in xy−plane bounded by y = x2

and y = 1

(c) tetrahedron in the first octant bounded by x+ z = 1 and y + 2z = 2

(d) solid in the first octant bounded by the surface x2 + y + z = 4

(e) solid in the first octant between the planes x+ y + 2z = 2 and 2x+ 2y + z = 4

7

LabPop11a # 3:Which of the following gives the volume of the cylinder x2 + y2 = 4 cut by the two planesz = 0 and x+ z = 3.

a.

∫ 2−2

∫ √4−x2−√4−x2

∫ 3−x0

dzdydx

b.

∫ 20

∫ √4−x2−√4−x2

∫ 3−x0

dzdydx

c.

∫ 2−2

∫ √4−x2−√4−x2

∫ 3−x0

(3− x)dzdydx

d.

∫ 20

∫ √4−x2−√4−x2

∫ 3−x0

(x2 + y2)dzdydx

e. None of these.

8

5. Set up a triple integral in cylindrical coordinates to find the volume of each solid:

(a) solid bounded above by the paraboloid z = 1 + x2 + y2, below by the xy−plane andon the sides by x2 + (y − 1)2 = 1

(b) solid bounded below by the cone z =√x2 + y2 and above by the paraboloid z =

2− x2 − y2

(c) solid whose base is the region in polar coordinates between r = cos θ and r = 2 cos θand the top lies in the plane z = 3− y

(d) prism whose base is a triangle in the xy−plane bounded by the x−axis, y = x andx = 1 and the top lies in the plane z = 2− y

9

LabPop11a # 4:

Evaluate:

∫ 2π0

∫ 20

∫ r20

zr dzdrdθ

a. 26π/3

b. 32π/3

c. 81π/5

d. 83π/5

e. None of these.

LabPop11a # 5:Which of the following gives the volume of the cylinder x2 + y2 = 4 cut by the two planesz = 0 and x+ z = 3.

a.

∫ 2π0

∫ 20

∫ 3−r cos θ0

r dzdrdθ

b.

∫ π0

∫ 20

∫ 3−r cos θ0

r dzdrdθ

c.

∫ 2π0

∫ 20

∫ 3−r cos θ0

r(3− r cos θ)dzdrdθ

d.

∫ π0

∫ 20

∫ 3−r cos θ0

r3dzdrdθ

e. None of these.

10

6. Evaluate the triple integral of f(x, y, z) = z over a solid enclosed by the paraboloid

z = x2 + y2 and the plane z = 4.

11