Math 181 Exam I Past Exam Questions
1) A horse trough has a isosceles trapezoidal cross section with a height of 2 ft and horizontal sides 2 ft
(bottom) and 4 feet (top). Assume that the length of the trough is 5 ft.
a) Compute the work required to pump the water out of the trough if it is full. You must first set
up a Riemann sum.
b) Set up, but do not compute an equation involving an integral to compute the fluid pressure
against one end of the plate if it is full.
2) Set up, but do not compute, an integral to find the area of the surface obtained by rotating
y  x 2 ,0  x  2 about y = -3
using a) dx b) dy
3) A 400-gallon tank initially contains 100 gallons of brine (salt water) containing 50 lb of salt. Brine
containing 1 lb of salt per gallon enters the tank at the rate of 5 gallons per second and the wellmixed brine in the tank flows out at the rate of 3 gallons per second. How much salt will the tank
contain when it is full?
4) Use the Newton’s Law of Cooling to solve the following: A metal plate that has been heated cools
from 180 degrees to 150 degrees in 20 minutes when surrounded by air at a temperature of 60
degrees. When will the temperature be 100 degrees? You must first solve the DE.
5) A) Find the centroid of the region in the first quadrant bounded by
y  a2  x2
B) Use the theorem of Pappus to find the volume of the solid obtained by rotating the region in the
y  a2  x2
first quadrant bounded by
about the y-axis.
6) Integrate each of the following :
a)
 x sec
7) Integrate a)
8) Integrate
9)
x
e
3x
x dx b)
6x  7
 ( x  2) 2 dx c)
1
3
x  25
2
dx
b)
 tan
dx
 e x  e x
 /2
d)
 sin
4
x cos 3 xdx
0
2
3
x dx

c)
16  x 2 dx
0
1
dx
 ex
b
e3x
lim
x   ln x
b)
4
 sin 2 x dx
b)
Compute a)
10) Integrate a)
2
lim (1  ax) x
x 0
3
3
 sec x tan x dx
c)

x2
x 1
dx
11) A) Set up, but do not compute, an integral to find the volume of the solid obtained by rotating the
region bounded by
x  y2  y  2
and
y
1
x
2
about x = 5. B) Set up, but do not compute, an integral
to compute the volume of the solid obtained by rotating the region bounded by (1,0), (2,2), (4,0) about
y = -1 using a) shell b) washer.
12) Integrate
13) a) Show
x 3  3x 2  2 x  1
dx

x3 1
 sec
3
x dx 
1
(sec x tan x  ln sec x  tan x )  c
2
b) Integrate
 sec
4
x dx
14) Find the volume of the solid whose base is the disk
x 2  y 2  25 and whose cross-sections
perpendicular to the x-axis are squares.
15) A) A semicircular plate 2 ft in diameter sticks straight down into fresh water with diameter along the
surface. Find the force exerted by the water on one side of the plate. You must first set up a
Riemann sum. B) ) A triangular plate shown below is submerged 2 ft into fresh water. Find the force
exerted by the water on one side of the plate. You must first set up a Riemann sum.
16) A cone-shaped water reservoir is 10 feet in radius across the top and 15 feet deep. If the reservoir is
filled to a depth of 10 ft, how much work is required to pump the water to the top of the reservoir?
You must first set up a Riemann sum.
17) Integrate a)
1

x  8 x  25
2
b)
dx

xe x dx
18) (5 points each)
Integrate a)
1
 x log
3
x
dx
b)

ln x
dx
x2
c)
 sec
4
x tan 2 x dx
19) A ship’s anchor, weighing 1000 lb, is attached to a 20 ft chain that weighs 200 lb. Find the work
required to pull the anchor up 20 ft. First set up a Riemann sum for the chain.
20) A)Set up, but do no compute, an integral that represents the area of the surface obtained by rotating
the curve
y
x , 0  x  2 about the x-axis using a) dx b) dy
B) Compute the arc length for
21) Solve
f ( x) 
e x  ex
, [0, ln 3]
2
dP
 kP and answer the following question: A certain cell culture has a doubling
dt
time of 5 hours. Initially, there were 3000 cells present. Find the time it takes for the
culture to triple?
22) Use Trig Sub to integrate
x
2
2
dx
 6x  5
23) Set up integrals to find the volume of the solid obtained by rotating the region in the first quadrant
bounded by
y  x and y  x 2 about a) x  3
b)
four integrals)
24) Compute a)
25) Integrate a)
2 

1  
lim
x 
x  
2x2  3
 x( x  1)2 dx
4x
b)
1
x 0 
b)
1
lim ( x  sin x )
x
2
tan 1 x dx
y  3 using both shell and washer. (you need to set up
26) A solid has, as its base, the circular region in the xy-plane bounded by
x2  y2  4 .
Find the volume
of the solid if every cross section perpendicular to the x-axis is an equilateral triangle with one side
in the base.
27) A circular plate of radius 2 feet is submerged vertically in water. If the distance from the surface of
the water to the center of the plate is 6 feet, find the force exerted by the water on one side of the
plate. You must first set up Riemann Sum.
28) Integrate each of the following: a)
e
1  e dx
3x
x
b)
e2x
 e x  4 dx
29) Set up, but do not compute an integral to find the volume of the solid generated by revolving the
region bounded by
x  2 y  y2, x  0
about the y = -2.
30) A spring has a natural length of 8 in. An 800-lb force stretches the spring to 12 in. How much work is
done in stretching the spring from 8 in to 12 in?
31) Set up, but do not evaluate, definite integrals that give the center of mass of a plate of density
 ( y )  1  y covering the region bounded by y  x 2 , x  0 and y  4 .
tan 2 x
dx
sec x
32) Integrate a)

33) Integrate a)
e
2x
b)
cos 2 x dx
 (ln x)
b)
34) Set up integrals to compute
 tan
2
dx
4
x dx
M,Mx,M y
of the region bounded by
y  6  x 2 and y  3  2 x if its
density is 1+x2
35) A spherical tank measures 20 ft in diameter. It is half-filed with kerosene weighting
51.2 lb/ft 3 .
How
much work does it take to pump the kerosene to the top of the tank? First find the Riemann sum.
36) Set up, but do not compute, the integral to find the volume of the solid obtained by rotating the region
bounded by
y  4  x2
and
y  x2  4
about a)
x  3
b)
y5
37) Use calculus to find the volume of a pyramid with height 30 in and square base with sides 10 in if the
cross sections parallel to the base are squares.
38) Find the length of the curve
y3
1
x

3 4y
from y = 1 to y = 3.