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EXAM 3 REVIEW SHEET
A. INTEGRATION – This includes finding the antiderivative, Fundamental Theorem of Calculus,
Substitution and Integration by parts.
1.
8
 x ln( x)dx
2.
4.
8  4x  9x 2
dx

x
5.
5
5
9 xdx
0 x 2  1
x
 xe dx
8.  ( x 3  sec 2 ( x)) dx
1
 f ( x)dx where
2
8 x  7
f ( x)   2
x
 cos(
x)
dx
x

6.  (e 8 x  sin( 3x)) dx
0
9  x2
1 x dx
2
9.
3
4
10. Compute the integral
3.
0
2
7.
4 x
 x e dx
x 1
x 1
.
b
B. AREA BETWEEN CURVES  ( f ( x)  g ( x)) dx
a
1. Find the area of the region bounded by the graphs f ( x)  2  x 2 and g ( x)  x.
Hint: Find a and b by setting f(x) = g(x).
2. Find the area of the region bounded by x  4 , x  54 , y = sin(x), and y = cos(x).
C. INTEGRAL APPLICATIONS
1. Given that f ( x)  10 cos( x) and f ( 6 )  2, find f(x).
2. Find s(t) given that v(t) = 3t 2  6t  5.
3. The rate that SARS spreads is modeled by N (t )  50te0.0375t . Given that
N (0)  1500 find N(t).
D. AVERAGE VALUE
Find the average value for the following
 x
1. y  sin    1 on [0, π]
2
2. y  x  x  1 on [0, 2]
2
x2
3. y  3
on [1, 5]
x 1
4. The population of a certain community t years after 2000 is given by the function
e 0.2t
P(t ) 
million people. What was the average population from 2000 to 2010?
4  e 0.2t
E. MORE INTEGRALS
5  4x 3  2x 6
1. 
dx
x6


7
5
3

2.   e 7 x  dx
3.  5 x  x 2  2 x 2 dx
x

5.  x(5 x 2  3 x  4)dx 6.  ( x e  e x  5)dx
4.  (3 x 2  cos(7 x))dx
7. Given that g ' ( x)  2 x 
3
, x > 0 and g (1)  3, find g(x).
x4
F. MORE INTEGRATION USING SUBSTITUTION
x
(ln( x)) 4
dx
1.  x 2 ( x 3  5) 9 dx
2. 
3.  2
dx
x 1
x
5.
x
10 x 2  1dx
6.
4.  cos( x) sin(sin( x)) dx
 tan( bx)dx
G. MORE INETGARTION BY PARTS
1.
 x cos(8 x)dx
2.
 xe
x
2
dx
3.
x
7
ln( x)dx
4.

ln( x )
dx
x3
Applications of exponentials and logarithms (sections 4.3& 4.4)
1. A company introduces a new product on a trial run in a city. They advertise the product on TV and find
the percentage of people (P) who buy the product after t ads have been run is given by the function
100%
P (t ) 
.
1  24e 0.28t
a. What percentage of people by the product without seeing the ads?
b. What percentage of people by the product after 1 ad has been run?
c. What percentage of people by the product after 20 ads have been run?
d. Find the rate of change of P(t).
e. Find the inflection point for P(t).
2. A dose of a drug is injected into a patient. The drug amount decreases by 10% per hour, that is
dA
 0.1A, where A is the amount in cubic centimeters and t is time in hours.
dt
a. An initial dose of 3ccs of the drug is administered to the patient. Find the function that models the
behavior of the drug.
b. How much of the drug is left in the patient after 10 hours?
c. How long will it take for half the initial amount to remain in the patient?
3. The decay rate of zirconium-88 is 0.83% per day. What is the half-life?
4. How old is a skeleton if 75% of the carbon-14 remains in the skeleton. The half life of carbon-14 is
5750 years.
5. Find the critical point and the inflection point(s) for the function f ( x)  e

( x  70) 2
8
.
ANSWERS
A. INTEGRATION – This includes finding the antiderivative, Fundamental Theorem of Calculus,
Substitution and Integration by parts.
5
ex
x 9 ln | x | x 9
9x 2
C
1.
2.
3. 2 sin( x )  C
4. 8 ln | x | 4 x 

C
C
5
9
81
2
9 ln( 26)
e 8 19
5.
6.
7. e2
8. tan(3) – 81/4 9. 9ln(2) – 3/2
10. 42

2
8
24
B. AREA BETWEEN CURVES
1. 9/2
2. 2 2
C. INTEGRAL APPLICATIONS
1. f ( x)  10 sin( x)  3
2. s(t )  t 3  3t 2  5t  s 0
D. AVERAGE VALUE
2
1. 1 
2. 4/3

E. MORE INTEGRALS
1
2
1.  5  2  2 x  C
x
x
1
4. x 3  sin( 7 x)  C
7
1
7. f ( x)  x 2  3  1
x
3.
1
12
ln( 63) 4. 12 ln
3. N (t ) 
4000 0.0375t 320000 0.0375t 333500
te

e

3
9
9
   0.411607 million or 411,607
4e 2
5
3
9
e7x
10 2 2 2
4
2.
3.
 5 ln x  C
x  x 
C
7
3
9
x
5
x e 1
5. x 4  x 3  2 x 2  C 6.
 e x  5x  C
4
e 1
F. MORE INTEGRATION USING SUBSTITUTION
(ln( x)) 5
ln( x 2  1)
( x 3  5)10
1.
2.
3.
C
C
C
5
2
30
4.  cos(sin( x))  C
5.
3
2
(10 x  1)
C
30
2
6.  ln cos( x)  C
G. MORE INTEGRATION BY PARTS
x
x
x sin( 8 x) cos(8 x)
2
2

C
1.
2. 2 xe  4e  C
8
64
ln( x)
1
x 8 ln( x) x 8
 2 C

C
3.
4. 
2
8
64
2x
4x
Sections 4.3, 4.4
ln( b) ln( 24)
x

 11.35
672e 0.28t
1. a. 4% b. 5.22% c. 91.85% d. P (t ) 
e.
k
0.28
 0.28t 2
(1  24e
)
y  L2  50
2. a. A(t )  3e 0.1t cc b. 1.1 cc c. 6.93 hours
3. 83.5 days
1
1
5. critical point (70, 1); inflection points (68, e 2 ) and (72, e 2 ).
4. 2386 years