MATH 152 Spring 2018COMMON EXAM II - VERSION A

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INSTRUCTOR:

SECTION NUMBER:

UIN:

DIRECTIONS:

1. The use of a calculator, laptop or cell phone is prohibited.

2. TURN OFF cell phones and put them away. If a cell phone is seen during the exam, your examwill be collected and you will receive a zero.

3. In Part 1 (Problems 1-15), mark the correct choice on your ScanTron using a No. 2 pencil. TheScanTron will not be returned, therefore for your own records, also record your choices on yourexam! Each problem is worth 4 points.

4. In Part 2 (Problems 16-20), present your solutions in the space provided. Show all your work neatlyand concisely and clearly indicate your final answer. You will be graded not merely on the finalanswer, but also on the quality and correctness of the work leading up to it.

5. Be sure to write your name, section number and version letter of the exam on the ScanTron form.

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Question Points Awarded Points

1-15 60

16 11

17 10

18 7

19 6

20 6

Total 100

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PART I: Multiple Choice. 4 points each.

1. Which of the following integrals gives the surface area obtained by rotating the curve x = e2t,y = tet, 0 ≤ t ≤ 1, about the y-axis?

(a)

∫ 1

02πe2t

√1

4e4t + (et + tet)2 dt

(b)

∫ 1

02πtet

√1

4e4t + (et + tet)2 dt

(c)

∫ 1

02πe2t

√e2t + (et + tet)2 dt

(d)

∫ 1

02πe2t

√4e4t + (et + tet)2 dt

(e)

∫ 1

02πtet

√4e4t + (et + tet)2 dt

2. After applying the appropriate substitution, which of the following is equivalent to

∫ 1/4

0

√1− 4x2 dx?

(a)

∫ π/6

0cos θ dθ

(b)1

2

∫ π/6

0cos2 θ dθ

(c)1

2

∫ π/3

0cos2 θ dθ

(d)

∫ π/3

0cos θ dθ

(e)

∫ π/6

0cos2 θ dθ

3.

∫ 4

3

x+ 4

x2 − 2xdx =

(a) 2 ln 4− 3 ln 2− 2 ln 3

(b) 2 ln 4 + 3 ln 2 + 2 ln 3

(c) −2 ln 4 + 3 ln 2

(d) −2 ln 4 + 3 ln 2 + 2 ln 3

(e) −2 ln 4 + 3 ln 2− 2 ln 3

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4. Find the tangent line to the parametric curve x = ln(t) + 4, y = 2t+ t2 at the point where t = 1.

(a) y = 4x− 4

(b) y = 4x− 13

(c) y =1

4x+ 4

(d) y =1

4x+

13

4(e) y = 4x− 16

5. Which of the following is a polar equation of the circle x2 + y2 = 2x?

(a) r = 2 cos θ

(b) r = 2

(c) r2 = 2 cos θ

(d) r = 2 sin θ

(e) r2 = 2 sin θ

6. Find the area of the part of the circle r = 4 sin θ that lies within the sector 0 ≤ θ ≤ π

3.

(a)4π

3−√

3

(b)8π

3− 2√

3

(c)4π

3+√

3

(d)4π

3− 1

(e)4π

3+ 1

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7.

∫ ∞

1

e1/x

x2dx =

(a) 0

(b) e− 1

(c) e

(d) 1− e(e) diverges

8. The improper integral

∫ ∞

1

sin(x) + 3

x2dx

(a) Diverges because

∫ ∞

1

3

x2dx diverges and

sin(x) + 3

x2≥ 3

x2

(b) Converges because

∫ ∞

1

2

x2dx converges and

sin(x) + 3

x2≤ 2

x2

(c) Diverges because

∫ ∞

1

4

x2dx diverges and

sin(x) + 3

x2≥ 4

x2

(d) Converges because

∫ ∞

1

4

x2dx converges and

sin(x) + 3

x2≤ 4

x2

(e) Converges because

∫ ∞

1

1

x2dx converges and

sin(x) + 3

x2≤ 1

x2

9. At which of the following points does x = 2t3 + 3t2 − 12t, y = t2 − 4t have a vertical tangent?

(a) (4,−4) and (13, 5) only

(b) (−7,−3) and (20, 12) only

(c) (−7,−3), and (0, 0) only

(d) (4,−4), (13, 5) and (0, 0) only

(e) (4,−4) only

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10.

∫ 1

0

x2 + 3x+ 4

x+ 1dx =

(a)3

2− 2 ln 2

(b) 4

(c)5

2+ 2 ln 2

(d)5

2− 2 ln 2

(e)3

2+ ln 2

11. Which of the following is the appropriate trigonometric substitution for

∫dx√

x2 + 6x+ 13?

(a) x+ 3 =√

13 tan θ

(b) x2 + 6x =√

13 sec θ

(c) x+ 3 = 4 tan θ

(d) x+ 3 = 2 sin θ

(e) x+ 3 = 2 tan θ

12. Which of the following integrals is/are improper?

I.

∫ 2

0

1

2x− 1dx II.

∫ ∞

1

1

x3dx III.

∫ 2

1x ln(2− x) dx

(a) I. II. and III.

(b) I. and II. only

(c) II. only

(d) II. and III. only

(e) I. and III. only

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13. The curve x = 1 + cos t, y = −2 + sin t

(a) Is a circle centered at (−1, 2) with radius 1, oriented clockwise.

(b) Is a circle centered at (−1, 2) with radius 1, oriented counterclockwise.

(c) Is a circle centered at (1,−2) with radius 1, oriented counterclockwise.

(d) Is a circle centered at (1,−2) with radius 1, oriented clockwise.

(e) None of these

14. Which of the following integrals gives the arc length of the cardioid r = 1 + sin θ?

(a)

∫ π

0

√2 dθ

(b)

∫ 2π

0

√2 dθ

(c)

∫ 2π

0

√1 + 2 sin θ dθ

(d)

∫ π

0

√1 + 2 sin θ dθ

(e)

∫ 2π

0

√2 + 2 sin θ dθ

15. The point (−1,√

3) is equivalent to which of the following in polar coordinates?

(a)

(−2,

5π

6

)(b)

(−2,

2π

3

)(c)

(2,

4π

3

)(d)

(2,

5π

6

)(e)

(−2,

5π

3

)

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PART II: Work Out: Box your final answer!

16. (a) (5 pts) Find the general form of the partial fraction decomposition of f(x) =3x3 + 6x+ 9

x4 + 3x2, and

find the numercial values of the coefficients.

(b) (6 pts) Using the result of part a.), find

∫3x3 + 6x+ 9

x4 + 3x2dx

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17. (10 pts) Find

∫dx

x4√x2 − 4

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18. (7 pts) Find the length of the curve x = 2e2t − t, y = 4et 0 ≤ t ≤ 3.

19. (6 pts) Consider the polar region R inside the circle r = 6 cos θ and outside the cardioid

r = 2 + 2 cos θ, as shown below. Set up but do not evaluate an integral that gives the area of R.

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20. (6 pts) Write the letter of the correct graph corresponding to the polar curve. No work needs to beshown.

r = 2 cos(4θ)

r = 2 sin(3θ)

A B C

D E F

G H I

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