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)
12' tla;rd NdKavhla i|yd jQ mßfNdacl b,a¨ï Y%s;h
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@
13. fudag¾ r: ghr ksIamdok iud.ula weia;fïka;= lr we;s wdldrh wkqj .ekqïlrejka úiska tallhl ñ, remsh,a D(q)= -0.1q2 + 90 jk úg ghr q ^oyia& m%udKhla b,a¨ï lrk w;r wdh;khtu m%udKh imhkafka S(q) = 0.2q2 + q + 50. ñ,g h'
i. iu;=,s; ñ, yd m%udKh .Kkh lrkak'
ii. iu;=,s;fhaoS mßfNdacl w;sßla;h yd ksIamdol w;sßla;h .Kkh lrkak'
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