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1
Revision Notes
Application of Integrals
Definite Integration
1. Definition:
If ( ) F x is an antiderivative of ( ) f x , then ( ) ( ) - F b F a is known as the definite integral of
( ) fx from a to b , such that the variable x, takes any two independent values say a and b .
This is also denoted as
b
a
f(x)dx
?
.
Thus ( ) ( )
b
a
f(x)dx F b F a =-
?
, The numbers a and b are called the limits of integration; a is
the lower limit and b is the upper limit.
2. Properties of Definite Integrals:
I.
ba
ab
f(x)dx= f(x) -
??
II.
ba
ab
f(x)dx= f(y)dy
??
III.
b c b
a a c
f(x)dx= f(x)dx+ f(x)dx
? ? ?
, where c may or may not lie between a and b .
IV .
aa
00
f(x)dx= f(a x)dx -
??
Page 2
1
Revision Notes
Application of Integrals
Definite Integration
1. Definition:
If ( ) F x is an antiderivative of ( ) f x , then ( ) ( ) - F b F a is known as the definite integral of
( ) fx from a to b , such that the variable x, takes any two independent values say a and b .
This is also denoted as
b
a
f(x)dx
?
.
Thus ( ) ( )
b
a
f(x)dx F b F a =-
?
, The numbers a and b are called the limits of integration; a is
the lower limit and b is the upper limit.
2. Properties of Definite Integrals:
I.
ba
ab
f(x)dx= f(x) -
??
II.
ba
ab
f(x)dx= f(y)dy
??
III.
b c b
a a c
f(x)dx= f(x)dx+ f(x)dx
? ? ?
, where c may or may not lie between a and b .
IV .
aa
00
f(x)dx= f(a x)dx -
??
2
V .
bb
aa
f(x)dx= f(a+b x)dx -
??
Note:
a.
a
0
f(x) a
dx=
f(x)+f(a x) 2 -
?
b.
b
a
f(x) b a
dx=
f(x)+f(a+b x) 2
-
-
?
VI.
2a a a
0 0 0
f(x)dx= f(x)dx+ f(2a x)dx -
? ? ?
a
0
0 if f(2a x) f(x)
=
2 f(x)dx if f(2a x) f(x)
- = - ??
??
??
-=
??
??
?
VII.
a
a
0
-a
2 f(x)dx if f( x) f(x)i.e.f(x)iseven
f
(x)dx
0 if f( x) f( x)i.e.f(x)isodd
??
-=
??
=
??
??
- = -
??
?
?
VIII. If f(x) is a periodic function of period (Tex translation failed) , i.e. f(a x) f(x) += , then
a.
na a
00
f(x)dx n f(x)dx =
??
b. ( )
na a
00
f(x)dx n 1 f(x)dx =-
??
Page 3
1
Revision Notes
Application of Integrals
Definite Integration
1. Definition:
If ( ) F x is an antiderivative of ( ) f x , then ( ) ( ) - F b F a is known as the definite integral of
( ) fx from a to b , such that the variable x, takes any two independent values say a and b .
This is also denoted as
b
a
f(x)dx
?
.
Thus ( ) ( )
b
a
f(x)dx F b F a =-
?
, The numbers a and b are called the limits of integration; a is
the lower limit and b is the upper limit.
2. Properties of Definite Integrals:
I.
ba
ab
f(x)dx= f(x) -
??
II.
ba
ab
f(x)dx= f(y)dy
??
III.
b c b
a a c
f(x)dx= f(x)dx+ f(x)dx
? ? ?
, where c may or may not lie between a and b .
IV .
aa
00
f(x)dx= f(a x)dx -
??
2
V .
bb
aa
f(x)dx= f(a+b x)dx -
??
Note:
a.
a
0
f(x) a
dx=
f(x)+f(a x) 2 -
?
b.
b
a
f(x) b a
dx=
f(x)+f(a+b x) 2
-
-
?
VI.
2a a a
0 0 0
f(x)dx= f(x)dx+ f(2a x)dx -
? ? ?
a
0
0 if f(2a x) f(x)
=
2 f(x)dx if f(2a x) f(x)
- = - ??
??
??
-=
??
??
?
VII.
a
a
0
-a
2 f(x)dx if f( x) f(x)i.e.f(x)iseven
f
(x)dx
0 if f( x) f( x)i.e.f(x)isodd
??
-=
??
=
??
??
- = -
??
?
?
VIII. If f(x) is a periodic function of period (Tex translation failed) , i.e. f(a x) f(x) += , then
a.
na a
00
f(x)dx n f(x)dx =
??
b. ( )
na a
00
f(x)dx n 1 f(x)dx =-
??
3
c.
b+na b
na 0
f(x)dx f(x)dx =
??
, where bR ?
d.
b+a
b
f(x)dx
?
independent of b .
e.
b+na a
b0
f(x)dx n f(x)dx =
??
, where n1 ?
IX. If f(x) 0 ? on the interval ? ?
a,b , then
b
a
f(x)dx 0 ?
?
.
X. If f(x) g(x) ? on the interval ? ?
a,b , then
bb
aa
f(x)dx g(x)dx ?
??
XI.
bb
aa
f(x)dx f(x) dx ?
??
XII. If f(x) is continuous on ? ?
a,b , m is the least and M is the greatest value of f(x) on
? ?
a,b , then
( )
b
a
m b a f(x)dx M(b a) - ? ? -
?
XIII. For any two functions f(x) and g(x) , integral on the interval ? ?
a,b , the Schwarz-
Bunyakovsky inequality holds
b b b
22
a a a
f(x).g(x)dx f (x)dx. g (x)dx ?
? ? ?
Page 4
1
Revision Notes
Application of Integrals
Definite Integration
1. Definition:
If ( ) F x is an antiderivative of ( ) f x , then ( ) ( ) - F b F a is known as the definite integral of
( ) fx from a to b , such that the variable x, takes any two independent values say a and b .
This is also denoted as
b
a
f(x)dx
?
.
Thus ( ) ( )
b
a
f(x)dx F b F a =-
?
, The numbers a and b are called the limits of integration; a is
the lower limit and b is the upper limit.
2. Properties of Definite Integrals:
I.
ba
ab
f(x)dx= f(x) -
??
II.
ba
ab
f(x)dx= f(y)dy
??
III.
b c b
a a c
f(x)dx= f(x)dx+ f(x)dx
? ? ?
, where c may or may not lie between a and b .
IV .
aa
00
f(x)dx= f(a x)dx -
??
2
V .
bb
aa
f(x)dx= f(a+b x)dx -
??
Note:
a.
a
0
f(x) a
dx=
f(x)+f(a x) 2 -
?
b.
b
a
f(x) b a
dx=
f(x)+f(a+b x) 2
-
-
?
VI.
2a a a
0 0 0
f(x)dx= f(x)dx+ f(2a x)dx -
? ? ?
a
0
0 if f(2a x) f(x)
=
2 f(x)dx if f(2a x) f(x)
- = - ??
??
??
-=
??
??
?
VII.
a
a
0
-a
2 f(x)dx if f( x) f(x)i.e.f(x)iseven
f
(x)dx
0 if f( x) f( x)i.e.f(x)isodd
??
-=
??
=
??
??
- = -
??
?
?
VIII. If f(x) is a periodic function of period (Tex translation failed) , i.e. f(a x) f(x) += , then
a.
na a
00
f(x)dx n f(x)dx =
??
b. ( )
na a
00
f(x)dx n 1 f(x)dx =-
??
3
c.
b+na b
na 0
f(x)dx f(x)dx =
??
, where bR ?
d.
b+a
b
f(x)dx
?
independent of b .
e.
b+na a
b0
f(x)dx n f(x)dx =
??
, where n1 ?
IX. If f(x) 0 ? on the interval ? ?
a,b , then
b
a
f(x)dx 0 ?
?
.
X. If f(x) g(x) ? on the interval ? ?
a,b , then
bb
aa
f(x)dx g(x)dx ?
??
XI.
bb
aa
f(x)dx f(x) dx ?
??
XII. If f(x) is continuous on ? ?
a,b , m is the least and M is the greatest value of f(x) on
? ?
a,b , then
( )
b
a
m b a f(x)dx M(b a) - ? ? -
?
XIII. For any two functions f(x) and g(x) , integral on the interval ? ?
a,b , the Schwarz-
Bunyakovsky inequality holds
b b b
22
a a a
f(x).g(x)dx f (x)dx. g (x)dx ?
? ? ?
4
XIV . If a function f(x) is continuous on the interval ? ?
a,b , then there exists a point ( ) c a,b ?
such that
b
a
f(x)dx f(c)(b a) =-
?
, where a c b ?? .
3. Differentiation Under Integral Sign:
Newton Leibnitz’s Theorem:
Given that x has two differentiable functions, g(x) and h(x) , where ? ?
x a,b ? and f is
continuous in that interval, then
? ? ? ? ? ? ? ?
h(x)
g(x)
d d d
f(t)dt h(x) .f h(x) g(x) .f g(x)
dx dx dx
??
=-
??
??
??
?
4. Definite Integral as a Limit of Sum:
Let f(x) be a continuous real valued function defined on the closed interval ? ?
a,b which is
divided into n parts as shown in figure.
Page 5
1
Revision Notes
Application of Integrals
Definite Integration
1. Definition:
If ( ) F x is an antiderivative of ( ) f x , then ( ) ( ) - F b F a is known as the definite integral of
( ) fx from a to b , such that the variable x, takes any two independent values say a and b .
This is also denoted as
b
a
f(x)dx
?
.
Thus ( ) ( )
b
a
f(x)dx F b F a =-
?
, The numbers a and b are called the limits of integration; a is
the lower limit and b is the upper limit.
2. Properties of Definite Integrals:
I.
ba
ab
f(x)dx= f(x) -
??
II.
ba
ab
f(x)dx= f(y)dy
??
III.
b c b
a a c
f(x)dx= f(x)dx+ f(x)dx
? ? ?
, where c may or may not lie between a and b .
IV .
aa
00
f(x)dx= f(a x)dx -
??
2
V .
bb
aa
f(x)dx= f(a+b x)dx -
??
Note:
a.
a
0
f(x) a
dx=
f(x)+f(a x) 2 -
?
b.
b
a
f(x) b a
dx=
f(x)+f(a+b x) 2
-
-
?
VI.
2a a a
0 0 0
f(x)dx= f(x)dx+ f(2a x)dx -
? ? ?
a
0
0 if f(2a x) f(x)
=
2 f(x)dx if f(2a x) f(x)
- = - ??
??
??
-=
??
??
?
VII.
a
a
0
-a
2 f(x)dx if f( x) f(x)i.e.f(x)iseven
f
(x)dx
0 if f( x) f( x)i.e.f(x)isodd
??
-=
??
=
??
??
- = -
??
?
?
VIII. If f(x) is a periodic function of period (Tex translation failed) , i.e. f(a x) f(x) += , then
a.
na a
00
f(x)dx n f(x)dx =
??
b. ( )
na a
00
f(x)dx n 1 f(x)dx =-
??
3
c.
b+na b
na 0
f(x)dx f(x)dx =
??
, where bR ?
d.
b+a
b
f(x)dx
?
independent of b .
e.
b+na a
b0
f(x)dx n f(x)dx =
??
, where n1 ?
IX. If f(x) 0 ? on the interval ? ?
a,b , then
b
a
f(x)dx 0 ?
?
.
X. If f(x) g(x) ? on the interval ? ?
a,b , then
bb
aa
f(x)dx g(x)dx ?
??
XI.
bb
aa
f(x)dx f(x) dx ?
??
XII. If f(x) is continuous on ? ?
a,b , m is the least and M is the greatest value of f(x) on
? ?
a,b , then
( )
b
a
m b a f(x)dx M(b a) - ? ? -
?
XIII. For any two functions f(x) and g(x) , integral on the interval ? ?
a,b , the Schwarz-
Bunyakovsky inequality holds
b b b
22
a a a
f(x).g(x)dx f (x)dx. g (x)dx ?
? ? ?
4
XIV . If a function f(x) is continuous on the interval ? ?
a,b , then there exists a point ( ) c a,b ?
such that
b
a
f(x)dx f(c)(b a) =-
?
, where a c b ?? .
3. Differentiation Under Integral Sign:
Newton Leibnitz’s Theorem:
Given that x has two differentiable functions, g(x) and h(x) , where ? ?
x a,b ? and f is
continuous in that interval, then
? ? ? ? ? ? ? ?
h(x)
g(x)
d d d
f(t)dt h(x) .f h(x) g(x) .f g(x)
dx dx dx
??
=-
??
??
??
?
4. Definite Integral as a Limit of Sum:
Let f(x) be a continuous real valued function defined on the closed interval ? ?
a,b which is
divided into n parts as shown in figure.
5
The point of division on x- axis are
( ) a,a h,a 2h.....a n 1 h,a nh, + + + - + where
ba
h
n
-
= .
Let
n
S denotes the area of these n rectangles.
Then, ( ) ( ) ( ) ( )
n
S hf(a) hf a h hf a 2h .... hf a n 1 h = + + + + + + + -
Clearly,
n
S is area very close to the area of the region bounded by
Curve y f(x), = x- axis and the ordinates xa = , xb = .
Hence
b
n
n
a
lim f(x)dx S
??
=
?
b
n-1
n
r0
a
fa lim (x)dx hf( +rh)
??
=
=
?
?
( )
n-1
n
r0
b
lim
ar
ba
fa
nn
??
=
?? -
- ??
=+
??
??
??
??
?
Note:
(a) We can also write
( ) ( ) ( )
n
S hf a h hf a 2h ..... hf a nh = + + + + + +
b
n
n
r1
a
b a b a
f(x)dx f a r
n
lim
n
??
=
?? -- ? ? ? ?
=+
? ? ? ? ??
? ? ? ?
??
?
?
(b) If
1
n1
n
r0
0
1r
a 0,b 1, f(x)dx f l
n
m
n
i
-
??
=
??
= = =
??
??
?
?
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FAQs on Revision Notes: Applications of Integrals - Mathematics (Maths) for JEE Main & Advanced
| 1. What are the applications of integrals in calculating areas under curves? |  |
Ans. Integrals are fundamentally used to compute the area under curves, which can be visualized as the region between the curve of a function and the x-axis over a specified interval. By using definite integrals, one can determine the exact area, which is essential in various fields such as physics, engineering, and economics. For example, if f(x) represents the height of a function, the area A under the curve from a to b is given by the integral A = ∫[a, b] f(x) dx.
| 2. How are integrals applied in physics, specifically in determining displacement and work? | |
Ans. In physics, integrals are used to calculate displacement when dealing with velocity functions. The relationship between displacement s, velocity v, and time t is expressed as s = ∫ v(t) dt, where the integral computes the total distance traveled over a time interval. Similarly, work done W by a variable force F can be calculated using the integral W = ∫[x₁, x₂] F(x) dx, which sums the infinitesimal work done over the distance from x₁ to x₂.
| 3. Can integrals be used in calculating volumes of solids of revolution? If so, how? | |
Ans. Yes, integrals play a crucial role in calculating the volumes of solids of revolution. This is typically achieved using the disk method or the washer method. For instance, if a region is revolved around an axis, the volume V can be computed using the integral V = π ∫[a, b] [f(x)]² dx for the disk method, where f(x) is the function defining the curve. This method allows for precise computation of volumes for various shapes formed by revolution.
| 4. What role do integrals play in probability and statistics? | |
Ans. In probability and statistics, integrals are used to find probabilities associated with continuous random variables. The probability density function (PDF) of a continuous random variable X can be integrated over an interval to find the probability that X falls within that interval. For example, P(a ≤ X ≤ b) = ∫[a, b] f(x) dx, where f(x) is the PDF. This application is vital in areas such as risk assessment and decision-making.
| 5. How can integrals be applied in calculating average value of a function over an interval? | |
Ans. The average value of a function f(x) over the interval [a, b] can be determined using integrals. The formula for the average value is given by Average = (1/(b - a)) ∫[a, b] f(x) dx. This formula calculates the mean value of the function within the specified interval, which is important in various applications, including economics for determining average cost or revenue.
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