Definite Integrals in Mathematics
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Learning Objectives
STUDENTS WILL BE ABLE TO KNOW
Definite integral.
Properties of definite integral.
To solve definite integral as the limit of a sum.
To solve definite integral using fundamental
theorem of integral calculus.
To solve definite integral using substitution
method.
Students will be able to solve different types of
definite integral problems.
They can apply it, to find the area bounded by
different curves.
Previous Knowledge
STUDENT SHOULD KNOW ABOUT :
Functions
Limits of a function
Derivatives
Continuous function
Indefinite Integrals
INTRODUCTION
Definite Integral:
The problem of finding area under a curve with in
a limit is called Definite Integral
.
The definite integral has a unique value.
A definite integral is denoted by , where a is
called the lower limit of the interval and b is called
the upper limit of the interval.
The Definite Integral is
introduced in two ways.
The Limit of a sum.
It has an anti derivative F in the interval
[a, b], then its value is the difference
between the value of F at the end points
i.e. F(b) – F(a).
1. Definite Integral as the limit of a sum :
Let f be a continuous function defined on
closed interval [a, b]. Assume that all
the values taken by the function are
non-negative, so the graph of the
function is a curve above the X-axis.
The Graph of
The Definite integral is the area bounded by the curve y
= f(x), the ordinates x = a, x = b and the X-axis.
In the figure,
Area of the rectangle ABLC ≤ Area of region ABDCA ≤
Area of the
rectangle
ABDM.
If s
n
and S
n
denote the sum of areas of
all lower rectangles and upper rectangles then,
Example:
Solution:
Here f(x) = x
2
+ 1 , a = 0 and b = 2, h = 2/n
⇒
nh = 2.
[1
[1
[1
[1
[1
[1
Video Link
Activity On Definite Integral
https://youtu.be/9lKDEJzgYY8
ASSIGNMENT
Evaluate the following definite integrals as limit of
sums :
2. Find Definite Integral, using Fundamental
Theorem of Integral Calculus.
First Fundamental Theorem of Integral Calculus :
Let f be a continuous function on the closed interval
[a, b] and A(x) be the area function,
i.e. A(x)
for all x
ϵ
[a, b].
Second Fundamental Theorem of Integral Calculus:
Let f be a continuous function defined on the closed
interval [a, b] and f be an anti derivatives of f, then,
Steps For Calculating
Example
:
Evaluate the following integral:
Solution
:
Evaluation of Definite Integral by
Substitution Method:
Steps to Evaluate
Consider the integral without limits and substitute,
y = f(x) or x = g(y) to reduce the given interval to a
known form.
Integrate the new integrand with respect to the new
variable without mentioning the constant of
integration.
Resubstitute for the new variable and write the
answer in terms of the original variable.
Find the value of the answers obtained in(3) at the
given limits of integral and find the difference of
the values at the upper and lower limits.
Example:
Evaluate
Properties of Definite Integrals
Properties of Definite Integrals
Example:
Evaluate
:
Example:
Evaluate
:
Example:
Evaluate
Example:
Evaluate:
1.
is equal to
1.
is equal to
Marking Scheme
(2 Mark)
Q:
Evaluate :
Solution :
We have,
Now
[1/2
[1/2
[1/2
[1/2
Marking Scheme
(
4 Mark)
Evaluate :
Solution :
Let
[using property
[1/2
[1/2
[1/2
[1
[1
[1/2
SUMMARY
We discussed the following points :-
Introduction of Definite Integral.
Definite Integral as limit of sum.
Definite integral using Fundamental
Theorem of Integral Calculus.
Definite Integral by Substitution
Method.
Properties of Definite Integrals.
MIND MAP
DEFINITE INTEGRALS
FORMULA FOR
LIMIT OF A SUM
PROPERTIES OF
DEFINITE
INTEGRALS
DIRECT
EVALUATION OF
DEFINITE
INTEGRALS
MODULUS OF
A FUNCTION
ODD AND
EVEN
FUNCTION
MIND MAP
WORKSHEETS
•
Basic Worksheet
•
Standard Worksheet
•
Advance Worksheet
Explore the concept of definite integrals in mathematics, covering properties, solving methods, and applications in finding areas bounded by curves. Prior knowledge of functions, limits, derivatives, and indefinite integrals is essential to grasp this topic effectively. Dive into the unique value of definite integrals, denoted by [a, b], and learn how to calculate it using the limit of a sum or the fundamental theorem of integral calculus. Enhance your problem-solving skills and apply definite integrals to various scenarios to find the bounded area accurately.
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Presentation Transcript
MATHEMATICS : STD. XII Class Topic : DEFINITE INTEGRAL
Learning Objectives STUDENTS WILL BE ABLE TO KNOW Definite integral. Properties of definite integral. To solve definite integral as the limit of a sum. To solve definite integral using fundamental theorem of integral calculus. To solve definite integral using substitution method. Students will be able to solve different types of definite integral problems. They can apply it, to find the area bounded by different curves.
Previous Knowledge STUDENT SHOULD KNOW ABOUT : Functions Limits of a function Derivatives Continuous function Indefinite Integrals
INTRODUCTION Definite Integral: The problem of finding area under a curve with in a limit is called Definite Integral. The definite integral has a unique value. A definite integral is denoted by , where a is called the lower limit of the interval and b is called the upper limit of the interval.
The Definite Integral is introduced in two ways. The Limit of a sum. It has an anti derivative F in the interval [a, b], then its value is the difference between the value of F at the end points i.e. F(b) F(a).
1. Definite Integral as the limit of a sum : Let f be a continuous function defined on closed interval [a, b]. Assume that all the values taken by the function are non-negative, so the graph of the function is a curve above the X-axis.
The Definite integral is the area bounded by the curve y = f(x), the ordinates x = a, x = b and the X-axis. In the figure, Area of the rectangle ABLC Area of region ABDCA Area of the rectangle ABDM. If sn and Sn denote the sum of areas of all lower rectangles and upper rectangles then,
Example: Solution: Here f(x) = x2 + 1 , a = 0 and b = 2, h = 2/n nh = 2. [1 [1 [1 [1 [1 [1
Video Link Activity On Definite Integral https://youtu.be/9lKDEJzgYY8
ASSIGNMENT Evaluate the following definite integrals as limit of sums :
2. Find Definite Integral, using Fundamental Theorem of Integral Calculus. First Fundamental Theorem of Integral Calculus : Let f be a continuous function on the closed interval [a, b] and A(x) be the area function, i.e. A(x) for all x [a, b]. Second Fundamental Theorem of Integral Calculus: Let f be a continuous function defined on the closed interval [a, b] and f be an anti derivatives of f, then,
Example: Evaluate the following integral: Solution:
Evaluation of Definite Integral by Substitution Method: Steps to Evaluate Consider the integral without limits and substitute, y = f(x) or x = g(y) to reduce the given interval to a known form. Integrate the new integrand with respect to the new variable without mentioning the constant of integration. Resubstitute for the new variable and write the answer in terms of the original variable. Find the value of the answers obtained in(3) at the given limits of integral and find the difference of the values at the upper and lower limits.
Example: Evaluate
Example: Evaluate :
Example: Evaluate:
Example: Evaluate
1. 1. is equal to is equal to Example: Evaluate:
Marking Scheme(2 Mark) Q: Evaluate : Solution : We have, Now [1/2 [1/2 [1/2 [1/2
Marking Scheme(4 Mark) Evaluate : Solution : Let [1/2 [using property [1/2 [1/2
[1/2 [1 [1
SUMMARY We discussed the following points :- Introduction of Definite Integral. Definite Integral as limit of sum. Definite integral using Fundamental Theorem of Integral Calculus. Definite Integral by Substitution Method. Properties of Definite Integrals.
MIND MAP DEFINITE INTEGRALS DIRECT PROPERTIES OF DEFINITE INTEGRALS FORMULA FOR LIMIT OF A SUM EVALUATION OF DEFINITE INTEGRALS MODULUS OF A FUNCTION ODD AND EVEN FUNCTION
WORKSHEETS Basic Worksheet Standard Worksheet Advance Worksheet
