Calculus 2
Test File
Spring 2005
Test #1
1.) Without
your calculator, find the area between the curves f(x) = px - x2 and
g(x) = sin x.
2.) Without
your calculator, find the volume of revolution if the region bounded by f(x) = 2x
- x2 + 3 and the x-axis is revolved around the x-axis.
3.) Without
your calculator, use cylindrical shells to find the volume of the region
bounded by f(x) = 5x - x2 and the x-axis when it is revolved around
the y-axis.
4.) Without
your calculator, use disks (or washers) to find the volume if the region
bounded by y = x, y = 5x and y = 5 is revolved around the y-axis
5.) Set
up the integral (AND USE YOUR CALCULATOR TO EVALUATE IT) to find
the arc length of the curve y = sin 3x, x = 0 to x = 2p.
6.) Suppose
we have the following pyramid.
Suppose
the base is a 4 foot by 4 foot square and the height is 5 feet. Now suppose the top is cut off, parallel to
the base, so the remaining solid has a height of 3 feet.
Use
integration and cross-sections to find the volume of the resulting solid.
7.) Set
up the integral (AND USE YOUR CALCULATOR TO EVALUATE IT) to find
the surface area of the surface generated by revolving the curve y = sin x, x =
0 to x = p, around the line y = 4.
8.) Find
the area of the region bounded by the curves y = sin x, y = 4 sin2x
+ 1, x = 0 and y = 4- 2x. You may use
your calculator
9.) Suppose the
region bounded by the curves y = 2 - x2 and y = x2 is
revolved around the line x = 6. Use
cylindrical shells to find the volume of the resulting solid. You may use your calculator.
10.) Use disks
or washers to find the volume if the region bounded by y = cos x, y = - cos x,
x = -p/2 and x = p/2 is revolved around the line y = 3. You may use your calculator.
Test #2
1.) Consider
the function f(x) = 2cx5, where c is a constant.
a.) Find
the value of c so that f(x) is a probability density function on the interval
[0, 1].
b.) Find
the mean.
c.) Find
the median.
2.) A
rectangular plate that is 2 feet high and 3 feet wide is submerged in water
(density = 62.4 pounds/cu. ft.). Find
the force exerted on the plate if the top edge of the plate is 3 feet below the
surface.
3.) Consider
the full water tank shown below.
What
is the work done in pumping water of the tank to a height 5 feet above the top
of the tank?
Do the following integrals. You may not use your calculator.
4.) 
5.) 
6.) 
7.) 
8.) 
9.) 
Test #3
1.) A 6-inch rod has variable mass. If the mass is given by (x) = x2,
where x is the distance from the left hand end of the rod, find the center of
mass of the rod. Tell how far it is from
the left hand end of the rod.
2.) Set
up BUT
DO NOT EVALUATE the partial fraction decomposition for the
following.
Evaluate
the following integrals. All work must
be shown. No calculators!
3.) 
4.) 
5.) 
EXACT ANSWER!! 6.) 
7.) 
Use the integral formulas in
your book to work these three problems. Write the number of any formula you use.
8.)
9.) 
10.) 
Test #4
1.) Evaluate
the following limits.
a.) 
b.) 
2.) Evaluate
the following integrals.
a.) 
b.) 
3.) Determine
whether or not the following sequences converge. If it converges, identify the limit. If it does not converge, explain why.
a.) 
b.) an
=
4.) Determine
convergence or divergence for the following series. If the series converges, to what does it
converge. If it does not converge,
explain why.
a.) 
b.) 
5.) Use
the integral test to determine convergence or divergence for the following
series. Be sure to show whether or not
the integral test applies.
6.) Determine
convergence or divergence for the following series. Be sure to make your reasons clear.
a.) 
b.) 
c.) 
Test #5
1.) Determine
convergence or divergence for the given series.
If it converges, determine if the convergence is absolute or conditional.
2.) Determine
convergence or divergence for the given series.
3.) Determine
convergence (absolute or conditional) or divergence for the given series.
4.) Consider the following series.
a.) Find
the sum of the first 50 terms of the series.
Feel free to use your calculator.
b.) Determine
convergence or divergence for the series.
If it converges, determine if the convergence is absolute or conditional.
5.) For the
following power series, use the root test to find the interval of
convergence. Do not forget to check the
endpoints.
6.) Find
the Taylor Series for f(x) = ex-1 around c = 1.
7.) Use a
familiar power series to find a power series for the following function. Then find the interval of convergence. Include endpoints.
8.) Use
the first 4 terms of the Maclaurin series for ex (hint: 
) to approximate e0.1. Then, find an upper bound for the error of
this estimate. You may use the following
formula for the remainder.
9.) Find
the Maclaurin series for f(x) = tan-1x.
Test #6
1.) Use
the Maclaurin series 
to evaluate the
following limit. Do NOT use L'Hopital.
2.) Convert
the following parametric equations to a single equation in x and y.
x = sin t, y = cos2t
3.) Graph
the following set of parametric equations.
x = sin (t/2), y = cos (t/3)
4.) Graph
the following set of parametric equations.
Show the starting point, ending point and direction of motion.
x = t2 + 2, y =
3t - 3, -2 < t < 2
5.) Find
the point(s) of intersection of the two sets of parametric equations.
6.) Find
the equation of the tangent line to the curve given by x = 2 cos t + sin
(2t), y = 2 sin t + cos (2t) at the
point (2, 1).
7.) Find
dy/dx for x = t2 - t, y = t4 - 4t2. Then find all points at which the curve has a
horizontal tangent or a vertical tangent.
Be sure to label your answers so I know what represents what.
8.) SET UP BUT DO NOT EVALUATE the
integral for finding the arc length for x = 2 cos t + sin (2t), y = 2 sin t + cos (2t) for 0 < t <
2p.
9.) SET UP BUT DO NOT EVALUATE the
integral for finding the surface area for the half-ellipse 
when it is revolved
around the x-axis. (Hint: Keep in mind that this half-ellipse can be
graphed using x = 3 cos t, y = 2 sin t, 0 < t < .)
10.) Now
evaluate ONE of the integrals from #8 or #9.
Be sure to tell me which one you did.
You may use your calculator to evaluate it.
Test #7
1.) a.) Convert (3,
-3) to polar coordinates. Give the EXACT ANSWER.
b.) Convert (-3, 5p/6) to rectangular
coordinates. Give the EXACT ANSWER.
2.) Convert
r = 2 cos q - sin q to rectangular coordinates and graph it.
3.) Graph
r = sin(q/2) cos(q/3).
4.) Find
the slope of the tangent line to the three-leaf rose r = sin 3qat q = 0 and q = p/4. EXACT
ANSWERS ONLY.
5.) Find
the area that is inside BOTH of the curves r = 2 and r = 3 + 2 cos q. Give the EXACT
ANSWER.
6.) Graph
the following ellipse. Fill in all
blanks. If a blank does not apply, write
"DNA."
Focus_______________ Focus__________________
Endpoint Minor Axis_______________ Endpoint Minor Axis________________
Vertex__________________ Vertex__________________
Asymptote________________ Asymptote________________
Center__________________
7.) Graph
the following hyperbola. Fill in all
blanks. If a blank does not apply, write
"DNA."
Focus_______________ Focus__________________
Endpoint Minor Axis_______________ Endpoint Minor Axis________________
Vertex__________________ Vertex__________________
Asymptote________________ Asymptote________________
Center__________________
8.) Identify
and graph the following conic.
9.) Identify
the kind of curve for each of the following equations.
a.) 2x2
+ 2y2 + Cx + Dy + E = 0
b.) 2x2
+ 5y2 + Cx + Dy + E = 0
c.) 5x2
- 4y2 + Cx + Dy + E = 0
d.) y2
- x + Dy + E = 0
e.) Cx +
2y + E = 0
10.) Graph
the following rotated conic. Fill in all
the blanks. If a blank does not apply,
write "DNA." SHOW ALL WORK!
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Graph
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Rotate Coordinate System
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Regular Coordinate System
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Focus
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Focus
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Vertex
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Vertex
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Center
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Asymptote
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Asymptote
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Final Exam
1.) Find
the area between the curves y = x2 + x + 2 and y = 2x + 2.
2.) Consider
the region bounded by y = x2 and y = 4. Find the volume of the resulting solid if
this region is revolved around the line x = 3.
3.) Consider
the region bounded by y = x2 and y = 4. Find the volume of the resulting solid if
this region is revolved around the line y = 0.
4.) A
cylindrical tank (radius 6 ft., height 10 ft.) is filled with water. Find the work done in pumping all of the
water out through the top of the tank.
You may use the fact that water has a weight density of 62.4 lbs/ft3.
In #5-10, evaluate the given integral. Do not use your calculator.
5.) 
6.) 
7.) 
8.) 
9.) 
10.) 
11.) Determine
convergence or divergence for the sequence {an} with an =
tan-1n. If it converges,
state the limit.
12.) Determine
convergence or divergence for the sequence {an} with an =
. If it converges,
state the limit.
13.) Determine
convergence or divergence for the following series. Show all necessary work to justify your
answer.
14.) Determine
convergence or divergence for the following series. Show all necessary work to justify your answer.
15.) Determine
convergence or divergence for the following series. If the series converges, determine if the
convergence is conditional or absolute.
Show all necessary work to justify your answer.
16.) Evaluate
the limit.
17.) Evaluate
the limit.
18.) Evaluate
the integral.
19.) Use
integration or differentiation of a known power series to find a power series
for ln(1+x).
20.) Graph
the following conic and fill in the blanks.
For any blank that is not applicable to the particular conic, write
"DNA."
Center _____
Vertex
or vertices____________
Focus
or Foci____________ Asymptote(s)______________
End
pts. Minor Axis__________________ _