The University of MemphisMATH 1920 Summer 2011Calculus II Dwiggins

FINAL EXAM

There are 12 questions worth ten points each.

# 1. Find the following antiderivatives:

.

Part I – Techniques of Integration

# 2. Calculate the following definite integrals:

(a) 3 2xx e dx

(b) 2 3sin cosx x dx

(b) 3

0

4 tan sec dxx x

(a)1

2

01 x dx

.

Calculus II Page 2Final Exam

# 3. Find the following antiderivatives:

(a) 2

2 1

4

xdx

x

Part II – Geometric Calculations

(b) 2

2

3 2

xdx

x x

# 4. (a) Sketch the region D bounded by the x-axis, the curve

(b) Show that two integrals are required to calculate the area of D using vertical strips, while only one integral is required if horizontal strips are used to partition D.

, and the line 6 .y x y x

(c) Calculate the area of D using whichever method is easiest.

223Answer

Page 3

# 6. Calculate the centroid of D, and use a theorem of Pappus to calculate the volume obtained when D is revolved about the line x = 4.

Calculus II Final Exam

# 5. Let D represent the region bounded by the y-axis, the line y = 4, and the curve x = y2.Calculate the volume obtained when D is revolved about:

(a) the x-axis

(b) the y-axis

Page 4Calculus II Final Exam

3/223# 7. (a) Calculate the arclength along the curve , 0 3.y x x

(b) Let C denote the curve y = x2 , 0 < x < 1.

Calculate the surface area obtained by revolving C about the y-axis.

Part III – Series and Approximation

# 8. Determine the interval of convergence for the power series0

( 3).

2 1

n

n

x

n

What happens when 4?

What happens when 2?

What happens when 0?

x

x

x

Page 5Calculus II Final Exam

# 9. Give the power series (with x0 = 0) for each of the following functions,

and give the interval of convergence for each series.

(a) 1

1 x(b) (c)xe 2cos( )x

20.5

0 (d) Use a series remainder theorem to calculate the value of

to within three decimal places.

(I got 0.4615 0.0005)

xe dx

Part IV – Analytic Geometry

# 10. Classify each of the following as the equation for an ellipse, a parabola, or a hyperbola.

2 2 1x y (a)

2 236 9y x (b)

1xy (c)

1

1 sinr

(d)

(e)10

5 4cosr

Page 6Calculus II Final Exam

# 11. Sketch the graph of the polar curve 4 2cosr and calculate the area bounded by this curve.

# 12. Consider the curve given parametrically by 3 23 , 3 .x t t y t

Fill in the following chart of values, indicating where the tangent line is either vertical or horizontal, and use this information to sketch the trajectory.

t x y dy/dx

–2

–1

0

1

2

What are the slopes of the tangent lines at the point where the curve crosses itself?

x y

3

3

x

y

Show that the second derivative tells us this curve is concave up for –1 < t < 1 and concave down otherwise.