Test Review
§ 4.1
1. Know the power rule for integration.
n 1
x
n
 x dx  n  1  C , n   1
a. ∫ (x 4 + x + x ½ + 1 + x – ½ + x – 2) dx =
5
2
3
1
x
x
2x 2
1
2


 x  2x  x  c
5
2
3
Remember you may differentiate to check your work!
1
Test Review
§ 4.1
2. Know the three steps in an application problem.
A $20,000 art collection is increasing in value at the
rate of 300√x dollars per year after x years.
Step 1 Find a formula for its value after t years.
We need the integral of f ‘ (x) or V = ∫ 300 x 1/2 dx
V = 300 ∫ x 1/2 dx = 200 x 3/2 + C
Step 2 Find the value of C.
Note we are given the value of $20,000 when x = 0.
20,000 = (200) (0 3/2 ) + C so 20,000 = C and
V = 200 x 3/2 + 20,000
Remember you may differentiate to check your work!
2
Test Review
§ 4.1
2. Know the three steps in an application problem.
A $20,000 art collection is increasing in value at the
rate of 300√x dollars per year after x years.
V = f (x) = 200 x 3/2 + 20,000
Step 3 Find the value in 25 years.
We need f (25) so
f (25) = (200) 25 3/2 + 20,000 =
200 · 125 + 20,000 = $45,000
3
Test Review
§ 4.2
1. Know the exponential rule for integration.
ax
e
ax
e
 dx  a  C
3x
e
e
dx
Find 

Find
 5e
1
 x
2
5e

1
x
2
1

2
3x
3
C
dx  5  e
1
 x
2
 c   10e
1
 x
2
dx 
c
Remember you may differentiate to check your work!
4
Test Review
§ 4.2
2. Know the logarithmic rule for integration.
1
 x dx  ln x  c
3
1
Find  dx  3  dx
x
x
 3 ln x  C
Remember you may differentiate to check your work!
5
4.3
General Indefinite Integral Formulas.
n 1
∫
un
u
du =
 C, n   1
n 1
∫ e u du = e u + C
1
 u du  ln u  C
Note the chain “du” is present!
6
4.3 Integration by Substitution.
Is it “Power Rule”, “Exponential Rule” or the “Log Rule”?
Step 1. Select a substitution that appears to simplify
the integrand. Use the basic forms in making your
choice. Make sure that du is a factor of the integrand.
Step 2. Express the integrand entirely in terms of u
and du, completely eliminating the original variable.
Step 3. Evaluate the new integral.
Step 4. Express the antiderivative found in step 3 in
terms of the original variable. (Reverse the
substitution.)
Remember you may differentiate to check your work!
7
4.3 Integration by Substitution
Examples
1. ∫ (3x + 5)4 dx
Let u = 3x + 5 and then du = 3 dx
1 4
1u
(3x  5)
1
4
u du 
c
c 
∫ (3x + 5) 3 dx 
3
3
15
3 5

2.  (x  1)e
x2  2x
5
5
dx Let u = x 2 + 2x and then du = 2x + 2 dx
1 u
1 u
1 x
1
x2  2x
2 (x  1)e
dx   e du  e  c  e

2
2
2
2
Remember you may differentiate to check your work!
2
 2x
c
8
4.3 Integration by Substitution
Examples
6
3. 
dx  6
2x  1
1
6
2
1
 2x  1 dx 
2
1
 2x  1 dx  6 2
Let u = 2x – 1 then
du = 2 dx
1
 u du  3 ln u  c
= 3 ln | 2x – 1| + c
Remember you may differentiate to check your work!
9
Test Review
§ 4.4
1. Know the basics of definite integrals.
Get out your calculator and turn it on!

3
1
x 2 dx

3
1
ln ( 2x  1) dx
10
Test Review
§ 4.5
1. Know the Average Value of a Continuous Function f
over [a, b].

b
a
f (x) dx
ba
Don’t forget to divide by b – a!
11
Average Value Problem
The temperature at time t hours is T(t) = - 0.3t 2 + 4t + 60
(for 0  t  12). Find the average temperature between
time 0 and 10.

b
a
f (x) dx
ba


10
0
(  0.3t 2  4t  60) dt
10

12
4.5 SUMMARY OF AREA PROBLEMS
The Area Between Two Curves.
Graph y = abs [f (x) – g (x)] in the interval of
integration from a to b. In some cases you may need to
use minimum to find the interval of integration.
That’s it!
Find the area between y = 3 – 2x 2
and y = 2x 2 – 4.

b
a
abs[f (x)  g (x)] dx
1. Graph
f (x) = abs [(3 – 2x 2 ) – (2x 2 – 4)]
2. Find the two x-intercepts by using minimum at
1.32 and - 1.32
3. Integrate over that domain.
.
Test Review
§ 4.6
1. Know how to calculate the Consumers’ Surplus.
Consumers' Surplus 
  d( x )  d( A ) dx
A
0
For the demand function d (x) = 200 e – 0.01x, find the
consumers’ surplus at a demand level of x = 100.
A is given as 100 and the market price
d (A) = d (100) = 200 e (– 0.01)(100) = 73.58

100
0


 200e0.01x - 73.58  dx  $ 5,284


The customers have paid $ 5,284 less than they were
willing to pay. A “savings”.
15
Test Review
§ 4.6
1. Know how to calculate a Gini index.
The Lorenz curve for the distribution of income for
students at York College is given by f (x) = x 1.5.
Find the index of income concentration.
Index 

1
0

1
0
2[x  L(x) ] dx
2(x  x 1.5 )dx
Know what the
answer means.
16
Don’t Forget
The cost, revenue, price, and profit formulas.
The average cost, average revenue, average price,
and average profit formulas.
The marginal cost, marginal revenue, marginal
price, and marginal profit formulas.
The marginal average cost, marginal average
revenue, marginal average price, and marginal
average profit formulas.