Showing posts with label Definite integrals. Show all posts
Showing posts with label Definite integrals. Show all posts
Sunday, January 17, 2016
27. Definite Integrals - Revision Facilitator
Revision Facilitator
Try to recollect relevant points on the topic.
If required right click on the topic and click on open in a new window to read the relevant material.
Close the window and come back.
1 The definite integral
2 Evaluation of definite integrals
3 Geometric interpretation of definite integral
4 Evaluation of integrals by substitution
5 Properties of definite integrals
6 Integral function
7 Summation of series using definite integral as the limit of a sum
8 Gamma function
Monday, November 17, 2008
The definite Integral - Definition
If F(x) is the antiderivative of a function f(x) continuous on (a,b)which means F'(x) = f(x) (a is less than x is less than b), then
∫f(x)dx {from a to b} = F(x) from a to b = F(b) - F(a)
∫f(x)dx {from a to b} = F(x) from a to b = F(b) - F(a)
Evluation of Definite Integrals
To find ∫f(x)dx from a to b
Find indefinite integral of ∫f(x)dx = ф(x)
Evaluate ф(b) and ф(a)
Calculate ф(b) - ф(a)
Find indefinite integral of ∫f(x)dx = ф(x)
Evaluate ф(b) and ф(a)
Calculate ф(b) - ф(a)
Geometric interpretation of definite integral
the definite integral represents the algebraic sum of the areas of the figures bounded by
the graph of the function y = f(x)
the x axis
the straight line x =a and x = b.
if the curve goes above and below the x axis in the interval a to b, the areas of above the x axis enter this sum with a plus sign, while those below the x axis enter it with a minus sign.
the graph of the function y = f(x)
the x axis
the straight line x =a and x = b.
if the curve goes above and below the x axis in the interval a to b, the areas of above the x axis enter this sum with a plus sign, while those below the x axis enter it with a minus sign.
Evaluation of Definite Integrals by Substitution
If a substitution is done in a definite integral, such a substitution needs to be effected at three places.
1. in the integrand
2. in the differential
3. in the limits
1. in the integrand
2. in the differential
3. in the limits
Properties of definite integrals
Properties of definite integrals
1. ∫ab f(x)dx = -∫ba f(x)dx
2. ∫ab f(x)dx = ∫ac f(x)dx + ∫cb f(x)dx
3. ∫0a f(x)dx = ∫0a f(a-x)dx
4. If f(-x) = f(x) (means f is an even function), then
∫-aa f(x)dx = 2∫0a f(x)dx
5. If f(-x) = -f(x) (means f is an odd function), then
∫-aa f(x)dx = 0
6. ∫0af(x)dx = ∫0af(a-x)dx and
∫abf(x)dx = ∫abf(a+b-x)dx
7. ∫0af(x)dx = ∫0a/2f(x)dx+∫0a/2f(a-x)dx
Due to the above relation
∫0af(x)dx = 0 if f(a-x) = -f(x)
∫0af(x)dx = 2∫0a/2f(x)dx if f(a-x) = f(x)
8. If f is continuous on [a,b], then the integral function defined by g(x) = ∫axf(t)dt for x Є [a,b]is derivable on [a,b], and g'(x) = f(x) for x Є [a,b].
9. If f(x0 is periodic with period T then
∫abf(x)dx = ∫a+nTb+nTf(x)dx, where n is an integer.
In particular
∫0nTf(x)dx = n∫0Tf(x)dx
If m and M are the smallest and greatest values of a function f(x) on an interval [a,b], then m(b-a)≤∫abf(x)dx≤M(b-a)
1. ∫ab f(x)dx = -∫ba f(x)dx
2. ∫ab f(x)dx = ∫ac f(x)dx + ∫cb f(x)dx
3. ∫0a f(x)dx = ∫0a f(a-x)dx
4. If f(-x) = f(x) (means f is an even function), then
∫-aa f(x)dx = 2∫0a f(x)dx
5. If f(-x) = -f(x) (means f is an odd function), then
∫-aa f(x)dx = 0
6. ∫0af(x)dx = ∫0af(a-x)dx and
∫abf(x)dx = ∫abf(a+b-x)dx
7. ∫0af(x)dx = ∫0a/2f(x)dx+∫0a/2f(a-x)dx
Due to the above relation
∫0af(x)dx = 0 if f(a-x) = -f(x)
∫0af(x)dx = 2∫0a/2f(x)dx if f(a-x) = f(x)
8. If f is continuous on [a,b], then the integral function defined by g(x) = ∫axf(t)dt for x Є [a,b]is derivable on [a,b], and g'(x) = f(x) for x Є [a,b].
9. If f(x0 is periodic with period T then
∫abf(x)dx = ∫a+nTb+nTf(x)dx, where n is an integer.
In particular
∫0nTf(x)dx = n∫0Tf(x)dx
If m and M are the smallest and greatest values of a function f(x) on an interval [a,b], then m(b-a)≤∫abf(x)dx≤M(b-a)
Integral Function
If f(x) is a continuous function defined on [a,b]m then a function ф(x) defined by
ф(x) = ∫f(t)dt, (a to x); x belongs to [a,b] is called the integral function of the function f.
ф(x) = ∫f(t)dt, (a to x); x belongs to [a,b] is called the integral function of the function f.
Summation of Series Using Definite Integral as the Limit of a Sum
∫f(x)dx (from a to b) = lim (h→0) h[f(a)+f(a+h)+f(a+2h)+...+f(a+(n-1)h)] where h = (b-a)/n
h→0 implies n→∞
∫f(x)dx (from 0 to 1) = lim (n→∞) (1/n)[Σf(r/n)(from r = to n-1)]
h→0 implies n→∞
∫f(x)dx (from 0 to 1) = lim (n→∞) (1/n)[Σf(r/n)(from r = to n-1)]
Gamma Function
If n is a positive number, then the improper integral
∫0∞ e-xxn-1dx is defined as a Gamma Function and is denoted by Γn.
∫0∞ e-xxn-1dx is defined as a Gamma Function and is denoted by Γn.
Friday, November 7, 2008
IIT JEE Mathematics Study Guide 27. Definite Integrals and Revision Facilitator
Sections in the chapter
1 The definite integral
2 Evaluation of definite integrals
3 Geometric interpretation of definite integral
4 Evaluation of integrals by substitution
5 Properties of definite integrals
6 Integral function
7 Summation of series using definite integral as the limit of a sum
8 Gamma function
Study Plans
Day 1
1 The definite integral
2 Evaluation of definite integrals
3 Geometric interpretation of definite integral
4 Evaluation of integrals by substitution
Day 2
5 Properties of definite integrals - up to example 15
Day 3
5 Properties of definite integrals - From example 16 to 30
Day 4
Illustrative Objective Type Examples 1 to 20
Day 5
6 Integral function
7 Summation of series using definite integral as the limit of a sum
8 Gamma function
Day 6
I.O.T.E.: 21 to 44
Day 7
Objective Type Exercises 1 to 20
Day 8
O.T.E.: 21 to 40
Day 9
O.T.E.: 41 to 60
Day 10
O.T.E.: 61 to 80
Day 11
O.T.E.: 81 to 100
Day 12
O.T.E.: 101 to 120
Day 13
O.T.E.: 121 to 140
Day 14
O.T.E.: 141 to 160
Day 15
O.T.E.: 161 to 180
Revision Period
Day 16
O.T.E.: 181 to 190
Day 17
O.T.E.: 191 to 200
Day 18
O.T.E.: 201 to 210
Day 19
O.T.E.: 211 to 220
Day 20
O.T.E.: 221 to 223
Fill in the blank type exercise 1 to 5
Day 21
Fill in the blank type exercise 6 to 15
Day 22
Fill in the blank type exercise 16 to 25
Day 23
Fill in the blank type exercise 26 to 35
Day 24
Fill in the blank type exercise 36 to 46
Day 25
True/False Type exercise 1 to 10
Day 26
True/False Type exercise 11 to 20
Day 27
True/False Type exercise 21 to 30
Day 28
True/False Type exercise 31 to 37
Day 29
Concept Revision
Day 30
Formula Revision
Revision Facilitator
Try to recollect relevant points on the topic.
If required right click on the topic and click on open in a new window to read the relevant material.
Close the window and come back.
1 The definite integral
2 Evaluation of definite integrals
3 Geometric interpretation of definite integral
4 Evaluation of integrals by substitution
5 Properties of definite integrals
6 Integral function
7 Summation of series using definite integral as the limit of a sum
8 Gamma function
Join IIT JEE Academy Orkut Community for more interaction
http://www.orkut.co.in/Main#Community.aspx?cmm=39291603
1 The definite integral
2 Evaluation of definite integrals
3 Geometric interpretation of definite integral
4 Evaluation of integrals by substitution
5 Properties of definite integrals
6 Integral function
7 Summation of series using definite integral as the limit of a sum
8 Gamma function
Study Plans
Day 1
1 The definite integral
2 Evaluation of definite integrals
3 Geometric interpretation of definite integral
4 Evaluation of integrals by substitution
Day 2
5 Properties of definite integrals - up to example 15
Day 3
5 Properties of definite integrals - From example 16 to 30
Day 4
Illustrative Objective Type Examples 1 to 20
Day 5
6 Integral function
7 Summation of series using definite integral as the limit of a sum
8 Gamma function
Day 6
I.O.T.E.: 21 to 44
Day 7
Objective Type Exercises 1 to 20
Day 8
O.T.E.: 21 to 40
Day 9
O.T.E.: 41 to 60
Day 10
O.T.E.: 61 to 80
Day 11
O.T.E.: 81 to 100
Day 12
O.T.E.: 101 to 120
Day 13
O.T.E.: 121 to 140
Day 14
O.T.E.: 141 to 160
Day 15
O.T.E.: 161 to 180
Revision Period
Day 16
O.T.E.: 181 to 190
Day 17
O.T.E.: 191 to 200
Day 18
O.T.E.: 201 to 210
Day 19
O.T.E.: 211 to 220
Day 20
O.T.E.: 221 to 223
Fill in the blank type exercise 1 to 5
Day 21
Fill in the blank type exercise 6 to 15
Day 22
Fill in the blank type exercise 16 to 25
Day 23
Fill in the blank type exercise 26 to 35
Day 24
Fill in the blank type exercise 36 to 46
Day 25
True/False Type exercise 1 to 10
Day 26
True/False Type exercise 11 to 20
Day 27
True/False Type exercise 21 to 30
Day 28
True/False Type exercise 31 to 37
Day 29
Concept Revision
Day 30
Formula Revision
Revision Facilitator
Try to recollect relevant points on the topic.
If required right click on the topic and click on open in a new window to read the relevant material.
Close the window and come back.
1 The definite integral
2 Evaluation of definite integrals
3 Geometric interpretation of definite integral
4 Evaluation of integrals by substitution
5 Properties of definite integrals
6 Integral function
7 Summation of series using definite integral as the limit of a sum
8 Gamma function
Join IIT JEE Academy Orkut Community for more interaction
http://www.orkut.co.in/Main#Community.aspx?cmm=39291603
Sunday, May 25, 2008
27 Definite Integrals - Revision points - 1
Properties of definite integrals
1. ∫ab f(x)dx = -∫ba f(x)dx
2. ∫ab f(x)dx = ∫ac f(x)dx + ∫cb f(x)dx
3. ∫0a f(x)dx = ∫0a f(a-x)dx
4. If f(-x) = f(x) (means f is an even function), then
∫-aa f(x)dx = 2∫0a f(x)dx
5. If f(-x) = -f(x) (means f is an odd function), then
∫-aa f(x)dx = 0
6. ∫0af(x)dx = ∫0af(a-x)dx and
∫abf(x)dx = ∫abf(a+b-x)dx
7. ∫0af(x)dx = ∫0a/2f(x)dx+∫0a/2f(a-x)dx
Due to the above relation
∫0af(x)dx = 0 if f(a-x) = -f(x)
∫0af(x)dx = 2∫0a/2f(x)dx if f(a-x) = f(x)
8. If f is continuous on [a,b], then the integral function defined by g(x) = ∫axf(t)dt for x Є [a,b]is derivable on [a,b], and g'(x) = f(x) for x Є [a,b].
9. If f(x0 is periodic with period T then
∫abf(x)dx = ∫a+nTb+nTf(x)dx, where n is an integer.
In particular
∫0nTf(x)dx = n∫0Tf(x)dx
If m and M are the smallest and greatest values of a function f(x) on an interval [a,b], then m(b-a)≤∫abf(x)dx≤M(b-a)
1. ∫ab f(x)dx = -∫ba f(x)dx
2. ∫ab f(x)dx = ∫ac f(x)dx + ∫cb f(x)dx
3. ∫0a f(x)dx = ∫0a f(a-x)dx
4. If f(-x) = f(x) (means f is an even function), then
∫-aa f(x)dx = 2∫0a f(x)dx
5. If f(-x) = -f(x) (means f is an odd function), then
∫-aa f(x)dx = 0
6. ∫0af(x)dx = ∫0af(a-x)dx and
∫abf(x)dx = ∫abf(a+b-x)dx
7. ∫0af(x)dx = ∫0a/2f(x)dx+∫0a/2f(a-x)dx
Due to the above relation
∫0af(x)dx = 0 if f(a-x) = -f(x)
∫0af(x)dx = 2∫0a/2f(x)dx if f(a-x) = f(x)
8. If f is continuous on [a,b], then the integral function defined by g(x) = ∫axf(t)dt for x Є [a,b]is derivable on [a,b], and g'(x) = f(x) for x Є [a,b].
9. If f(x0 is periodic with period T then
∫abf(x)dx = ∫a+nTb+nTf(x)dx, where n is an integer.
In particular
∫0nTf(x)dx = n∫0Tf(x)dx
If m and M are the smallest and greatest values of a function f(x) on an interval [a,b], then m(b-a)≤∫abf(x)dx≤M(b-a)
Sunday, April 27, 2008
Definite Integrals - Part 1
If F(x) is the antiderivative of a function f(x) continuous on (a,b)which means F'(x) = f(x) (a
∫f(x)dx {from a to b} = F(x) from a to b = F(b) - F(a)
Definite integral - Geometrical interpretation
the definite integral represents the algebraic sum of the areas of the figures bounded by
the graph of the function y = f(x)
the x axis
the straight line x =a and x = b.
if the curve goes above and below the x axis in the interval a to b, the areas of above the x axis enter this sum with a plus sign, while those below the x axis enter it with a minus sign.
Good online material
http://tutorial.math.lamar.edu/Classes/CalcI/DefnofDefiniteIntegral.aspx
∫f(x)dx {from a to b} = F(x) from a to b = F(b) - F(a)
Definite integral - Geometrical interpretation
the definite integral represents the algebraic sum of the areas of the figures bounded by
the graph of the function y = f(x)
the x axis
the straight line x =a and x = b.
if the curve goes above and below the x axis in the interval a to b, the areas of above the x axis enter this sum with a plus sign, while those below the x axis enter it with a minus sign.
Good online material
http://tutorial.math.lamar.edu/Classes/CalcI/DefnofDefiniteIntegral.aspx
Monday, October 22, 2007
Study Guide Ch.26. DEFINITE INTEGRALS
JEE Syllabus
definite integrals and their properties, application of the Fundamental Theorem of Integral Calculus.
Integration by parts, integration by the methods of substitution and partial fractions, application of definite integrals to the determination of areas involving simple curves.
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JEE question
If f″(x) = − f(x) and g(x) = f′(x) and F(x) = [f(x/2)]^2 + [g(x/2)]^2 and given that F(5) = 5 then F(10) is equal to
(A) 5
(B) 10
(C) 0
(D) 15
answer A
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definite integrals and their properties, application of the Fundamental Theorem of Integral Calculus.
Integration by parts, integration by the methods of substitution and partial fractions, application of definite integrals to the determination of areas involving simple curves.
-------------------
JEE question
If f″(x) = − f(x) and g(x) = f′(x) and F(x) = [f(x/2)]^2 + [g(x/2)]^2 and given that F(5) = 5 then F(10) is equal to
(A) 5
(B) 10
(C) 0
(D) 15
answer A
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