MATH 2433 (Section 20708)

TA: Jeric Alcala

Office: PGH 612

Email: jsalcala@uh.edu

LabPop10a # 1:

Which of the following gives the area of the region bounded by the line y = x+ 1 and parabola

y = x2 + 1?

a.

∫ 10

∫ x+1x2+1

dy dx

b.

∫ 10

∫ x2+1x+1

dy dx

c.

∫ 21

∫ √y−1y+1

dx dy

d.

∫ 21

∫ y−1√y−1

dx dy

e. None of these.

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1. Set up iterated integrals that gives the volumed of each of the following solids:

(a) tetrahedron bounded by the coordinate planes and the plane 2x+ 3y + 5z = 30

(b) upper hemisphere of x2 + y2 + z2 = 27

(c) under the paraboloid z = x2 + y2 but above the region on xy−plane bounded by

x = y2 and x = 4

(d) bounded by the cylinder x2 + y2 = 4 and planes z = 0 and z + y = 5

(e) on the first octant bounded by the plane z = 1− y and cylinder y =√x

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LabPop10b # 2:

Which of the following gives the volume of the tetrahedron in the first octant bounded by

the plane 2x+ y + 4z = 4?

a.

∫ 20

∫ 2−y/20

4− 2x− y4

dy dx

b.

∫ 20

∫ 4−2x0

4− 2x− y4

dy dx

c.

∫ 40

∫ 4−2x0

4− 2x− y4

dy dx

d.

∫ 40

∫ 2−y/20

4− 2x− y4

dy dx

e. None of these.

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2. Transform each into equivalent polar integral then evaluate.

(a)

∫ 4−4

∫ √16−x20

dy dx

(b)

∫ 40

∫ 0−√

16−y2dx dy

(c)

∫ 10

∫ √1−x2−√1−x2

ex2+y2dy dx

(d)

∫ 0−1

∫ 0−√1−x2

2

1 +√x2 + y2

dy dx

(e)

∫ √31

∫ x1

dy dx

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LabPop10b # 3:

Which of the following is equivalent to

∫ 0−2

∫ √4−x20

√1− x2 − y2 dy dx in polar coordi-

nates?

a.

∫ π/20

∫ 20

r√

1− r2dr dθ

b.

∫ ππ/2

∫ 20

r√

1− r2dr dθ

c.

∫ ππ/2

∫ 2−2r√

1− r2dr dθ

d.

∫ π/20

∫ 2−2r√

1− r2dr dθ

e. None of these.

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3. Set up a double integral in polar coordinates that gives the area of the following region:

(a) interior of the cardioid r = 2− 2 sin θ

(b) intersection of the interiors of r = 1 + cos θ and r = 1

(c) inside the loop of r = 1− 2 sin θ

(d) inside one leaf of r = 12 cos θ

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LabPop10b # 4:

Which of the following gives the area of the region outside circle r = 1 and inside the

circle r = 2 sin θ

a.

∫ π0

∫ 2 sin θ1

rdr dθ

b.

∫ 5π/6π/6

∫ 2 sin θ1

rdr dθ

c.

∫ 5π/3π/3

∫ 2 sin θ1

rdr dθ

d.

∫ π/6−π/6

∫ 12 sin θ

rdr dθ

e. None of these.

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4. 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

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LabPop10b # 5:

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.

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