The University of Memphis
MATH 1920 Summer 2011
Calculus II
Dwiggins
FINAL EXAM
There are 12 questions worth ten points each.
Part I – Techniques of Integration
# 1. Find the following antiderivatives:
(a)

(b)
 sin
x3e2 x dx
2
x cos3 x dx
# 2. Calculate the following definite integrals:

(a)
1
1  x 2 dx
0
.
(b)


0
4
tan 3 x sec x dx
Calculus II
Final Exam
Page 2
# 3. Find the following antiderivatives:
(a)

2x 1
dx
x2  4
.
(b)

x2
dx
x  3x  2
2
Part II – Geometric Calculations
# 4. (a) Sketch the region D bounded by the x-axis, the curve y  x , and the line y  6  x .
(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.
(c) Calculate the area of D using whichever method is easiest.
 Answer  22 3 
Calculus II
Final Exam
Page 3
# 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
# 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
Page 4
Final Exam
# 7. (a) Calculate the arclength along the curve y  23 x3/2 , 0  x  3.
(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 series
What happens when x  4?
What happens when x  2?
What happens when x  0?


n 0
( x  3) n
.
2n  1
Calculus II
Page 5
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)
e x
(c) cos( x 2 )
(d) Use a series remainder theorem to calculate the value of

0.5
0
e x dx
2
to within three decimal places.
(I got 0.4615  0.0005)
Part IV – Analytic Geometry
# 10. Classify each of the following as the equation for an ellipse, a parabola, or a hyperbola.
(a)
x2  y 2  1
(b)
y 2  36  9 x 2
(c)
xy  1
(d)
r
1
1  sin 
(e)
r
10
5  4 cos 
Calculus II
Page 6
Final Exam
# 11. Sketch the graph of the polar curve r  4  2cos
and calculate the area bounded by this curve.
3
2
# 12. Consider the curve given parametrically by x  t  3t , y  t  3 .
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
x
y
y
dy/dx
–2
 3
–1
0
1
3
2
What are the slopes of the tangent lines at
the point where the curve crosses itself?
Show that the second derivative tells us
this curve is concave up for –1 < t < 1
and concave down otherwise.
x