Chapter 7

Additional Integration TopicsSection 1

Area Between Curves

Area Between Two CurvesLet y = f(x) and y = g(x) be two continuous functions with f(x) > g(x) on [a,b]. Then the area between the graphs of the two curves on [a,b] is given by the definite integral ( ) ( )

b

af x g x dx

f(x)

g(x)

( ) ( )b

af x g x dx

ExampleArea Between Two Curves

2Find the area under the curve and above the curve between 0 and 1.

y xy x x x

Solution: First examine the graphs of the two functions.

0

0.2

0.4

0.6

0.8

1

0.2 0.4 0.6 0.8 1

y x

2y x

Find the area between the two curves and1xy

12 xy

Find the area of the region in the fourth quadrant bounded by the x-axis, y-axis, and 12 xy

Area of a Curve Under the x-axis

If the graph of y = f(x) is below the x-axis on [a,b], then the area below the x-axis and above the graph of y = f(x) on [a,b] is

area ( )b

af x dx

x

y Top: y = 0a b

Bottom: y = f(x)

top bottom

0 ( )

( )

b

a

b

a

b

a

A dx

f x dx

f x dx

Find the area enclosed between the curves and on

xy 22 xy 3 , 1

Read Example 3 page 413

Consumers’ and Producers’ Surplus

• There are consumers who are ready to pay a higher price for the commodity than the going price .

The total of all savings is called the consumers’ surplus.• There are producers who are willing to sell the commodity

for less than the going price . so the current price represents a savings. The total of all savings is called the producers’ surplus.

0p

0p

0x

0p

D(x)

S(x)

x

pAt the equilibrium point consumers will purchase the same number of the commodity as the producers will supply

Section 7.2

0

))((' 0

x

o

dxpxDSurplusConsumers

0

))((' 0

x

o

dxxSpSurplusproducers

point mequilibriu theis ,x 00 pwhere

Example

If the demand equation is given by and the supply equation is given by , find the consumer’s surplus.

218.045)( xxDp xxxSp 5.112.0)( 2

Continuous Money Flow (stream)

• When money is being compounded, we are not suggesting that the bank is actually continuously placing money into our account. Rather, we calculate what is in our account as if this were in fact happening. We can think of our account as a continuous flow of money.

If we know the rate of change of this flow (stream), call it ,we can obtain A(t) which is the total amount of income obtained by this flow of money, such that

)()( tftA

)(tf

Continuous Income StreamTotal Income for a Continuous Income Stream:

If f (t) is the rate of flow of a continuous income stream, the total income produced during the time period from t = a to t = b is

b

adttf )( income Total

a Total Income b

Example

Suppose an investment brings in an income of $1000 per year. How much actual income is obtained in 3 years?

Example

Suppose the rate of change (flow) of income in thousands of dollars per year from an oil well is projected to be , where t is in years. Find the total amount of income produced from the well during the first year.

tetf 1.0100)(

Example

Find the total income produced by a continuous income stream in the first 2 years if the rate of flow is

f (t) = 600 e 0.06t

275,1$100010

00010

600 income Total

120

20

060

2

0

06.0

) – (e,

e,

dte

.

t .

t

Future Valueof a Continuous Income

StreamFrom previous work we are familiar with the continuous compound interest formula

A = Pert.

If f (t) is the rate of flow of a continuous income stream, 0 t T, and if the income is continuously invested at a rate r compounded continuously, the the future value FV at the end of T years is given by

T trTrT tTr dtetfedtetfFV00

)( )()(

The difference between future value and total income is interest

ExampleLet’s continue the previous example where

f (t) = 600 e0.06 t

Find the future value in 2 years at a rate of 10%.

And how much is the interest earned?

Integration Using TablesSection 7.4

• General power rule

• Rules for e and for ln

• Substitution

• Integration by parts

These various techniques are helpful guidelines but can prove to be difficult at times. The most convenient technique may be to refer to a table of integrals

Previous sections have shown several methods for integration to include

Example

1Find (2 3)

dxx x

Solution:Using a table of integrals, we find the general integral of the form

1 1 ln for 0( )

xdx bx ax b b ax b

Example

2

1Find 9 1

dxx

Solution:

The general form that resembles the integrand now is2 2

2 2

1 lndx x x ax a

Substitution and Integral Tables

Sometimes the formula matches exactly, as in the preceding example.

Sometimes a substitution needs to be made in order to fit one of the formulas on the table.

Example

This almost fits formula 41:

If u = 3x , u2 = 9x2, du = 3dx and a = 1, we could make the necessary adjustments.

dxxx 19 22

2242222222 ln2

81 auuaauauuduauu

2

02

1dx

ex

Evaluate

x