Test Review

1

§ 4.1

1. Know the power rule for integration.n 1

n xx dx C, n 1

n 1

a. ∫ (x 4 + x + x ½ + 1 + x – ½ + x – 2) dx =

35 2 2 1 12x x 2xx 2x x c

5 2 3

Remember you may differentiate to check your work!

Test Review

2

§ 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.

200 x 3/2 + C

We need the integral of f ‘ (x) or V = ∫ 300 x 1/2 dx

Find a formula for its value after t years.Step 1

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

Step 2 Find the value of C.

Remember you may differentiate to check your work!

V = 300 ∫ x 1/2 dx =

Test Review

3

§ 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.

We need f (25) so f (25) = (200) 25 3/2 + 20,000 =

Find the value in 25 years.Step 3

We need f (25) so f (25) = (200) 25 3/2 + 20,000 = 200 · 125 + 20,000 = $45,000

V = f (x) = 200 x 3/2 + 20,000

Test Review

4

§ 4.2

1. Know the exponential rule for integration.ax

ax ee dx C

a

Find 3xe dx

3xeC

3

Find 1

x25e dx

1x

25 e dx

1

x 12 x2

5ec 10e c

12

Remember you may differentiate to check your work!

Test Review

5

§ 4.2

2. Know the logarithmic rule for integration.

Find3

dxx

3 ln x C

1dx ln x c

x

13 dx

x

Remember you may differentiate to check your work!

6

4.3General Indefinite Integral Formulas.

∫ un du = 1n,C1n

u 1n

∫ e u du = e u + C

Culnduu

1

Note the chain “du” is present!

7

4.3 Integration by Substitution. Is it “Power Rule”, “Exponential Rule” or the “Log Rule”?

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.)

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.

Remember you may differentiate to check your work!

4.3 Integration by Substitution Examples

8

1. ∫ (3x + 5)4 dx

41u du

3

Let u = x 2 + 2x and then du = 2x + 2 dx

u1e du

2

Let u = 3x + 5 and then du = 3 dx

∫ (3x + 5)4 3 dx 1

3

2x 2x2. (x 1)e dx2x 2x1

2 (x 1)e dx2

51 uc

3 5

5(3x 5)c

15

u1e c

2

2x 2x1e c

2

Remember you may differentiate to check your work!

4.3 Integration by Substitution Examples

9

63. dx

2x 1

Let u = 2x – 1 then du = 2 dx

= 3 ln | 2x – 1| + c

1 26 dx

2 2x 11 1

6 du2 u

3 ln u c

16 dx

2x 1

Remember you may differentiate to check your work!

Test Review

10

§ 4.4

1. Know the basics of definite integrals.

dxx3

1

2

dx)1x2(ln3

1

Get out your calculator and turn it on!

Test Review

11

§ 4.5

1. Know the Average Value of a Continuous Function f over [a, b].

Don’t forget to divide by b – a!

b

af (x) dx

b a

Average Value Problem

12

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 10 2

a 0f (x) dx ( 0.3t 4t 60) dt

b a 10

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.

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.

b

aabs[f (x) g (x)] dx

Test Review

15

§ 4.61. Know how to calculate the Consumers’ Surplus.

For the demand function d (x) = 200 e – 0.01x, find the consumers’ surplus at a demand level of x = 100.

-0.01x10

0

0200e 73.58 dx

The customers have paid $ 5,284 less than they were willing to pay. A “savings”.

A is given as 100 and the market price d (A) = d (100) = 200 e (– 0.01)(100) = 73.58

$ 5,284

0

AConsumers' Surplus d( x ) Ad( ) dx

Test Review

16

§ 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.

1 1.5

02(x x )dx

Know what the answer means.

1

0L(xIndex 2[ x)x ] d

Don’t ForgetThe 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.