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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
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.
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
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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
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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
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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
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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.