Integral Calculus

In differential calculus we differentiate a function to

obtain another function called derivative. Integral

calculus is concerned with the opposite process.

Reversing the process of differentiation and finding

the original function from the derivative is called

integration or anti-differentiation.

What is integral calculus?

The Indefinite Integral

If F(x) is a function of x and its derivative,

then F(x) is called an indefinite integral or simply an integral

of f(x). Symbolically this is written as,

)()()]([

xfxFdx

xFd

cxFdxxf )( )(

• The symbol ∫ is the integral sign, f(x) is integrand ^wkql,Hh& and c is the constant of integration.

• Constant c may have different values and accordingly we

can have different members of f(x)dx family.

For example, if y = x4, the derivative of y is 34xdx

dy

•When is given, to find y we have to follow the

opposite process of differentiation. Thus,

34xdx

dy

434 xdxx

For example it is equivalent to the derivatives of

y = x4+1, y = x4 - 5, y = x4 + a etc.

• Thus, if we add a constant (c) to the integral it will

match with the differential function.

34xdx

dy

cxdxx434

However, may be the derivative of different

differentiable functions of y.

Rules for Indefinite Integration

-1n ; 1

1

cn

xdxx

nn

Rule 1: the Power Rule

eg. cxdxx65

6

1

dxxfkdxxkf )()(

Rule 2. the Integral of a Multiple

eg. cxdxxdxx 433

2

122

; k is a constant

Rule 3.the Integral of a Sum or Difference

dxxgdxxfdxxgxf )()()()(

dxxdxdxxdxxx 352)352(22

cxxx 32

5

3

2 23

Rule 4. Generalized Power Function Rule

can

baxdxbax

nn

)1(

)()(

1

cxx

cx

xdxx

33

22

3323 )1(

9

1

33

)1()1(

cxx

dxx

2/3

2/32/1 )24(

6

1

42

3

)24()24(eg. i.

eg. ii.

01

x xlndxx

Rule 5. the Logarithmic Rule

cxdxx

dxx

ln31

33

c 12 ln

2

3

12

13

12

3xdx

xdx

x

i.

ii.

Rule 5. If f(x) is a function of x and its

differential coefficient with respect to x is

f ′(x). The derivative of ln[±f(x)] is

)(

)()](ln[

xf

xf

dx

xfd

)(ln)(

)( xfdx

xf

xf

eg. i.

1ln1

2 22

xdxx

x

dxx

x

13

2

eg. ii.

The derivative of the denominator is 3x2, and to apply the

above rule the numerator should be multiplied by 3 as such

denominator, too.

c1xln3

1

1

3

3

1

3)1(

3

3

3

2

3

2

dx

x

xdx

x

x

Rule 6: the Exponential Rule

cedxexx

cexxe

dxedxxdxedxexe

x

xx

24

1

23)23(

34

2424eg. i.

i.

cxf

edxe

xfxf

)(

)()(

ii.

eg. i.

cedxe xx 3434

4

1

iii. )(][

, )()(

)( xfedx

edeyWhen xf

xfxf

cedxxfe xfxf )()( )(

dxxex2

of integral indefinite Findeg. i.

f(x) = x2 and xxf 2)(

To apply the above rule we have to adjust the original

function as,

dxxex2

22

1 There is not a fundamentaldifference between the original

function and this function.

We can apply the above rule for this function

dxxex2 cedxxe xx

222

2

1

Rule 8: the Substitution rule

dxxx )1(22

Let u = x2+1; then du/dx = 2x or dx = du/2x.

Now du/2x can be substituted for dx of the above function.

cxx

cx

cu

)12(2

1

)1(2

1

2

1

24

22

2

udux

duxudxxx

22)1(2 2

eg. dxxx32 )8(2

If we defined u = x2+ 8 and du/dx = 2x. From this

x

duxudxxudxxx

222)8(2 3332

dxx

du

2

cx

uduu

42

43

)8(4

1

4

1

The Definite Integral

The indefinite integral of a continues function f(x) is:

cxFdxxf )()(

If we choose two values of x in the domain, say a and b

(b > a), substitute them successively into the right side of

the above equation and form the difference we get a

numerical value that is independent of the constant c.

)()()()( aFbFcaFcbF

This value is called the definite integral of f(x) from a to b.

a and b are lower and upper limits of integration,

respectively.

Evaluate the following definite integrals

56243

33x .1 334

2

34

2

2

xdx

2. 27)186(2

1

2 dxxx

)()(

)()()()(

aFbF

caFcbFcxFdxxfb

a

b

a

Now, we will modify the integration sign to indicate

the definite integral of f(x) from a to b as:

ba abb

axx eekkedxke )( .4

3.

1040(4160)4

1

9

xdxx

4

4

44

2

0

2

0

43

74

1

70724

1

4

77

The Definite Integral as an Area

A

a b X

Y

0

The area of the region

bounded by the curve y = f(x),

and by the x axis, on the left

by x = a, and on the right by

x = b is given by,

Area b

a

dxxfA )()(

y=f(x)

If the curve y = f(x) lies below the x axis, then

b

a

dxxfA )()(Area

The area (A) + B) = b

a

c

b

dxxfdxxf )()(

X0a b

Y

y = f(x)

c

A

B

IF f(x) and g(x) are the two functions of x and f(x) > g(x)

JA

B

y = f(x)

y = g(x)

0X

ba

Y

Area (A) = Area (J) –Area (B)

b

a

b

a

dxxgdxxf )()(

eg.1. Determine the area under the curve given by the

function y = 20 – 4x over the interval 0 to 5.

y = 20 – 4x

Y

Y50

20

A

Area (A) =

50

(0)-50-100

220)420(

5

0

5

0

2

xxdxx

Eg. 2 Find the area bounded by the functions

y = x2, and y = 10 – 3x and Y axis.

y = x2

y = 10 - 3x20

A

A = 2

0

2

0

2)310( dxxdxx

3

34

32

310

2

0

32

0

2

xxx

3. Determine the area between the curve of f(x) = 10 - 2X

and the X axis for values of X = 3 to X = 7.

e

0 X3 5 7

4a

b dc

f(x) = 10 – 2X

AREA = abc + bde

f(x)

10

-4

AREA abc =

4

]10[)210(

5

3

53

2

XXdXX

AREA bde =

4 -

]10[)210(

7

5

75

2

XXdXX

AREA = abc + bde = 8

1. If the marginal cost (MC) function of a firm is

and the fixed cost CF = 90. Find the total cost function

(TC).

QeC 2.02

2. If the marginal cost (MC) function of a firm is

and fixed cost of the

function is 1088. Find the total cost function (TC).

3642)46()( 32 qqqqMC

Economic Applications

3. If the marginal saving function of a country is

. If the aggregate saving S is zero

when income (Y) is 81. Find the saving function S(Y).

21

1.03.0)(

YYS

4. Consumer’s demand function for a given commodity

has been estimated to be P = 30 – 2Q

where, P is the price of a unit of the commodity and Q is

the per capita consumption of the commodity per person

per month. Determine (a) the total expenditure and (b) the

consumer surplus when the price of a unit is 5.

5. If the supply function of a commodity is P = 1000 + 50Q

where, P is the price per unit and Q is the number of units

sold each day. Find the producer surplus when the price of a

unit of the commodity is 2000.

6. If the willingness of a nurse to provide her service is

defined by the supply function W = 2.5 + 0.5H

where, W is the wage rate per unit

H is hours of work provided each week.

Determine the producer surplus paid to the nurse if the

prevailing wage rate is 9 per hour.

9. Suppose that t years from now, one investment will be generating

profit at the rate of hundred dollars per year, while a

second investment will be generating profit at the rate of

hundred dollars per year. P1(t) and P2(t),

satisfy P2(t) ≥ P1(t) for the first N years (0≤ t ≤ N).

(a) For how many years does the rate of profitability of the second

investment exceed that of the first?

(b) Compute the net excess profit for the time period determined in

part (a). Interpret the net excess profit as an area.

2'1 50)( ttP

ttP 5200)('2

10. Suppose that when it is t years old, a particular industrial

machine generates revenue at the rate R (t) = 5,000 - 20t2 dollars per

year and that operating and servicing costs related to the machine

accumulate at the rate C (t) 2,000 + 10t2 dollars per year

(a) How many years pass before the profitability of the machine

begins to decline?

(b) Compute the net earnings generated by the machine over the time

period determined in part (a)

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D(q) = 4(25-q2) fõ'

i. NdKavfhka tall 3la ñ,g .ekSu i|yd mßfNdaclhd f.ùug iQodkï uqo,a m%udKh fldmuK o@

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i. iu;=,s; ñ, yd m%udKh .Kkh lrkak'

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