Review Problems for Part II of the Final Exam in MA112
1. Consider the graph shown below.
4
f(t)
3
2
1
0
0.5
1
1.5
t
2
2.5
3
Suppose the area under the curve y = f (t) for t between 0 and x is given by the function
A(x) = 14 x2 + 1 − cos(x).
(a) Find the area under the curve y = f (t) on the interval [1, 2]. Give your answer to three (3)
decimal places.
(b) Describe the relationship between f (t) and A(x).
(c) Determine the function f (t).
2. Two lanes of a running√track are modeled by semiellipses. The interior lane√(lane 1) is modeled
by the equation y1 = 105 − 0.2x2 , while the outside lane (lane 2) is y2 = 150 − 0.2x2 . The
starting point for lane 1 is at the point (−20, 5). The finish points for both lanes are where
the line y = 5 intersects the curve in the first quadrant (x, y > 0). Where should the starting
point (both x and y coordinates!) be placed on lane 2 so that the two lane lengths will be equal
(running clockwise)? Give your answer to three (3) decimal places. Set up any integrals needed
to solve this problem
3. You are designing a new decorative vase. After much
“playing”, you come across a shape which
√ −x/2
you obtained by rotating the function f (x) = 2 xe
about the x-axis, between x = 1 and
x = 6, where x is measured in inches. Note that the opening of the vase occurs at x = 6.
(a) Set up and evaluate (not necessarily by hand) the integral needed to find the volume of your
vase. Give your answer to three (3) decimal places.
(b) If you are to stain this vase (except for the base) with some protectant that costs you 3 cents
per square inch to apply, how much does it cost you? Set up and evaluate (not necessarily
by hand) any integrals needed to solve this problem. Give your answer to three (3) decimal
places.
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4. A cup of cocoa is placed in a room that is at a constant temperature of 20◦ C. Initially, the temperature of the cocoa is 80◦ C. After 10 minutes, the temperature of the cocoa is 60◦ C. Determine
the temperature of the cocoa as a function of time. When is the temperature of the cocoa equal
to 37◦ C?
5. A tank with a volume of 100 liters initially contains 40 liters of brine and 80 kg of salt. Pure
water is pumped into the tank at a rate of 3 liters per minute, and the mixture is pumped out at
a rate of 2 liters per minute. How much salt is in the tank when the tank fills up?
6. Throughout this problem, let f (x) denote a continuous function about which we know only the
function values given in the table below.
x : 1.0
1.2
1.4 1.6 1.8 2.0
f (x) : 2.0 −1.5 −0.4 1.3 1.7 2.6
Z2
(a) Evaluate f 0 (x)dx, where f 0 (x) denotes the derivative of f (x).
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(b) Use the trapezoidal rule with n = 5 subintervals to estimate
Z2
f (x)dx.
1
7. Suppose that 27 foot-pounds of work is required to stretch a certain spring from its natural length
to 4 feet beyond its natural length. If a person starts pulling on the spring when it is already
stretched to 3 feet beyond its natural length and then stretches it another 5 feet beyond that,
how much work is required of the person? Assume Hooke’s law.
8. Set up an integral for the area bounded by the curves y = tan(x), y = x, and x = π/4.
(a) Find the value of c such that the line x = c divides the area from part (a) into two equal
parts.
9. The area under the graph of the function f (t) between t = 0 and t = x is equal to x3 + 9x. What
is f (1) equal to? Explain how you obtained your answer.
10. When a slice of the White Chapel forty-nine feet from the end is taken, the roof is in the shape
of two joined arcs of circles with a common radius of 41 feet. The arc on the right has angle
0 ≤ θ ≤ 0.955, in radians. Determine the area under the roof (the shaded region above).
=.955
r =41'
2
r =41'
11. Let f (x) =
1
(x−1)2/3
=
1
.Explain
|x−1|2/3
why
Z
2
f (x) dx is an improper integral. Carefully evaluate
0
the integral, showing all important steps. Hint: (−1)1/3 = −1.
12. (The Decay and Fall): Assume that acceleration due to gravity is a constant 9.80 meters per
second squared. The half-life of Carbon-10 is 19.45 seconds. A sphere with a total (constant) mass
of 2.000 kg and terminal velocity of 49.0 meters per second experiences air resistance proportional
to the sphere’s velocity. When the sphere contains 10.0000 grams of Carbon-10 (out of 2.000 kg
total mass) it is dropped from a height of 800 meters.
(a) Set up an initial value problem for the velocity v(t) of the sphere. Solve the resulting DE
for v(t), use v(t) to determine the height of the sphere above the ground as a function of t,
and use this to determine how long it takes the sphere to fall to the ground. Remember, the
total mass of the sphere is a constant 2.000 kg
(b) Determine how much Carbon-10 remains in the sphere at the moment that it hits the ground.
13. The graph of a function f (x) is shown below.
Note that the areas of regions bounded by y = f (x) and the x-axis are given on the graph.
Rx
Throughout this problem let g(x) denote the function g(x) = f (t)dt.
−2
(a) (a) Evaluate the definite integral
R3
f (x)dx
−2
3
(b) Evaluate the definite integral
−2
R
f (x)dx.
2
(c) Is g(x) increasing or decreasing at x = −1? Explain your answer.
(d) Let h(x) = g(x2 ). Estimate the value of h0 (−1) as closely as possible
14. Consider the region in the xy - plane bounded by x = y 2 − 4 and y = 2 − x. A sketch of this
region will be helpful in completing parts (a) and (b) below.
(a) Set up an integral or sum of integrals, using x as the variable of integration, that gives the
area of the region.
(b) Set up an integral or sum of integrals, using y as the variable of integration, that gives the
area of the region.
(c) Now evaluate the integrals you set up in parts (a) and (b) and confirm that they are equal.
15. (a) Find the volume of the solid obtained by revolving about the x-axis the region bounded by
y = 5 − x2 and y = 1. (Show the integral which gives this volume and evaluate it.)
(b) Use the trapezoidal rule to approximate the arclength of the graph of y = g(x), for 1 ≤
x ≤ 3, where g(x) denotes a continuous function about which we know only the values of its
DERIVATIVE g 0 (x) shown in the table below:
x:
1 4/3 5/3
2
7/3
8/3
3
q
√
√
√
√
√
(Show the integral which
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g (x) :
8
3
0 − 25 − 3 − 8 − 24
gives the arclength, and use the trapezoid rule to approximate it.)
0
16. The integral
−2
R
−∞
1
dx
x5
is a convergent improper integral.
Carefully rewrite this integral as an
appropriate limit, and then show the steps in evaluating this integral, as though you were doing
this task by hand. Confirm your result using Maple.
17. When water runs out through a small hole in the bottom of a barrel, the depth y (in inches) of
the water in the barrel at time t (in seconds) satisfies the differential equation
dy
√
= −k y
dt
where k is a positive constant.
(a) Verify by direct substitution into the differential equation, that
1
y = (C − kt)2
4
is a general solution to the differential equation for 0 ≤ t ≤
C
.
k
(b) If initially the water depth is 40 inches, and 30 seconds later it is 37 inches, how long will it
take for all the water to run out of the barrel?
4
18. It takes a force of 21, 714 lb. to compress a coil spring assembly on a NYC Transit Authority
subway car from its free height of 8 in. to a compressed height of 5 in. Find the assembly’s spring
constant, and then determine the work required to compress the assembly the first half inch from
its free height. (Assume Hooke’s Law, and answer to the nearest inch-pound.)
19. A tank of capacity 150 gallons initially contains 100 gallons of pure water. A brine solution of
concentration 0.5 lb. of salt per gallon enters the tank at a rate of 3 gallons per min., and the
well-mixed solution in the tank is pumped out at the same rate. Determine the amount of salt
in the tank at time t.
20. Find the total area enclosed by the graphs of y = x2 − 12 x + 1 and y = x3 + 2x2 − x + 1.
21. If a cup of hot chocolate (180 degrees) is placed in a room whose temperature is 70 degrees,
how long will it take for the cup to cool to 100 degrees? Use Newton’s Law of Cooling (i.e.
the rate of change of the temperature of the object is proportional to the difference between the
temperature of the object and its surroundings) and assume that the temperature after 5 minutes
is 140 degrees.
22. Let R be the region in the first quadrant whose boundaries are the x − axis, the y − axis, and
the graph of y = 2 − x2 . Find the negative number a so that the volume of the solid generated
by revolving R about the line x = a is 25 cubic units.
23. A 500 gallon tank contains a brine solution with the initial amount of salt being 300 pounds.
Suppose a brine solution consisting of 1/2 pound of salt per gallon enters the tank at a rate of 8
gallons per minute. If the mixture leaves at the same rate, determine the amount of salt in the
tank at time t.
24. A force of 200 Newtons stretches a spring 20 centimeters. How much work is done in stretching
the spring from 10 centimeters to 30 centimeters?
t
25. Suppose that the curve C is define parametrically by x = cos(t), y = e 5 sin(2t), 0 ≤ t ≤ 4π. Give
a sketch of C and determine its length.
26. Your company has an order to make a blown glass piece of art. The shape of the object is found
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by revolving the curve y = x( 2 ) (2 − x) , 0 ≤ x ≤ 2 π , about the x-axis where units are in feet.
The hollow piece of art is to be filled with water. How much water will the artwork hold? What
is the surface area of the blown glass piece of art?
27. Before a goose lands in a pond it circles the area. If the path of a certain goose is given by the
t
t
parametric equations x(t) = e(− 2 ) cos(t) , y(t) = e(− 2 ) sin(t) , how far has the goose flown for
0 ≤ t ≤ 2 π? Ignore any changes in elevation.
28. Jocelyn was supposed to be home before midnight. Her mother returned home at 1:30 AM. If the
temperature of the engine in Jocelyn’s car was 190 degrees F when her mother returned home, and
186 degree F twenty minutes later, when do you think Jocelyn returned home? Assume that the
running temperature of Jocelyn’s care is about 220 degrees F and that the outside temperature
is 82 degrees F.
5
29. A stuntwoman weighing 140 lbs (including gear) steps out of a plane at 5000 feet. It is known
that air resistance is proportional to velocity, and that for this stuntwoman the constant of
proportionality is 1.2 lbs/sec. with the chute closed. How fast will be stuntwoman be falling 20
seconds after she leaves the plane? How far will she have fallen 20 seconds after she leaves the
plane?
30. In order to find the volume of water that can pass through a channel, a team of flood control
specialists needs to determine the cross-sectional area of a river. The river is 200 feet wide.
Surveyors were hired to determine the depth of the water every 20 feet across the river. The
following data was obtained: (0,0), (20,17), (40,24), (60,36), (80,31), (100,28), (120, 30), (140,
18), (160, 9), (180, 3), (200,0). Use the trapezoidal rule to obtain an estimate for the crosssectional area of the river.
31. A region R in the first quadrant is bounded by the graphs of
√
3
y = 3 − 2x − x 2 + x2 and y = x.
(a) Sketch a graph of the region. Then set up a definite integral that gives the area of the region
R and find the value of that integral.
(b) Set up a definite integral that gives the volume of the solid of revolution obtained by revolving
R about the line y = 3. Then find the value of that integral.
32. Let f be a continuous function for which
Z2
f (x)dx = 15 and
0
If g(x) =
Zx
f (t)dt, find the value of
0
Z2
xf (x)dx = −4.
0
Z2
g(x)dx. Use integration by parts with u = g(x).
0
33. For a differentiable function f (x), some values of its derivative f 0 (x) are given in the table below.
x:
3.0
3.5 4.0 4.5 5.0
f (x) : −2.2 −0.4 0.3 1.2 1.5
0
Write out the definite integral that gives the length of the graph of f (x) from x = 3.0 to x = 5.0,
and then use the trapezoidal rule to approximate the value of that integral.
6
34. A contractor is required to pour a solid concrete ramp from ground level to a loading dock 4 feet
above ground level. Specifications require that the foot of the ramp be 12 feet from the dock,
measured horizontally along the ground, that the width of the ramp at ground level be 4 feet,
and that the width of the ramp at the dock level be 6 feet. A picture of the ramp is provided
below. Find the number of cubic feet of concrete needed to pour the ramp.
35. A tank initially contains 300 gallons of a salt water solution containing 50 pounds of salt. Suppose
that a solution of concentration 15 pound per gallon is pumped into the tank at a rate of 10 gallons
per minute and the well mixed solution is pumped out of the tank at the same rate. SET UP and
SOLVE the appropriate initial value problem in order to determine the function x(t) = number
of pounds of salt in the tank at time t minutes for all t ≥ 0.
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36. Throughout this problem let f (x) = sin(x3 − x + 1).
(a) Use Maple to find P4 (x), the fourth Taylor polynomial for f (x) centered about c = 0. Write
the result below with the coefficients written as decimal numbers rounded to three decimal places.
(b) Find
1/2
Z
P4 (x)dx where P4 (x) is the Taylor polynomial you found in part (a).
0
(c) The value of the integral you computed in part (b) should approximate
1/2
Z
f (x)dx. Graph
0
f (x) and P4 (x) for 0 ≤ x ≤ 1 on the same set of axes and copy the graph in the space below.
1/2
1/2
Z
Z
P4 (x)dx approximates
f (x)dx based on this graph and/or other
Comment on how well
0
0
observations.
37.
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