Tamil Nadu Board of Secondary EducationHSC Science Class 12th
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Tamil Nadu Board Samacheer Kalvi solutions for Class 12th Mathematics Volume 1 and 2 Answers Guide chapter 9 - Applications of Integration [Latest edition]

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Class 12th Mathematics Volume 1 and 2 Answers Guide - Shaalaa.com
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Chapter 9: Applications of Integration

Exercise 9.1Exercise 9.2Exercise 9.3Exercise 9.4Exercise 9.5Exercise 9.6Exercise 9.7Exercise 9.8Exercise 9.9Exercise 9.10
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Exercise 9.1 [Page 96]

Tamil Nadu Board Samacheer Kalvi solutions for Class 12th Mathematics Volume 1 and 2 Answers Guide Chapter 9 Applications of IntegrationExercise 9.1 [Page 96]

Exercise 9.1 | Q 1 | Page 96

Find an approximate value of `int_1^1.5` xdx by applying the left–end rule with the partition {1.1, 1.2, 1.3, 1.4, 1.5}

Exercise 9.1 | Q 2 | Page 96

Find an approximate value of `int_1^1.5` x2dx by applying the right–end rule with the partition {1.1, 1.2, 1.3, 1.4, 1.5}

Exercise 9.1 | Q 3 | Page 96

Find an approximate value of `int_1^1.5 (2 - x)` dx by applying the mid-point rule with the partition {1.1, 1.2, 1.3, 1.4, 1.5}

Exercise 9.2 [Page 98]

Tamil Nadu Board Samacheer Kalvi solutions for Class 12th Mathematics Volume 1 and 2 Answers Guide Chapter 9 Applications of IntegrationExercise 9.2 [Page 98]

Exercise 9.2 | Q 1. (i) | Page 98

Evaluate the following integrals as the limits of sum:

`int_0^1 (5x + 4)"d"x`

Exercise 9.2 | Q 1. (ii) | Page 98

Evaluate the following integrals as the limits of sum:

`int_1^2 (4x^2 - 1)"d"x`

Exercise 9.3 [Pages 112 - 113]

Tamil Nadu Board Samacheer Kalvi solutions for Class 12th Mathematics Volume 1 and 2 Answers Guide Chapter 9 Applications of IntegrationExercise 9.3 [Pages 112 - 113]

Exercise 9.3 | Q 1. (i) | Page 112

Evaluate the following definite integrals:

`int_3^4 (d"x)/(x^2 - 4)`

Exercise 9.3 | Q 1. (ii) | Page 112

Evaluate the following definite integrals:

`int_(-1)^1 ("d"x)/(x^2 + 2x + 5)`

Exercise 9.3 | Q 1. (iii) | Page 112

Evaluate the following definite integrals:

`int_0^1 sqrt((1 - x)/(1 + x)) "d"x`

Exercise 9.3 | Q 1. (iv) | Page 112

Evaluate the following definite integrals:

`int_0^(pi/2) "e"^x((1 + sin x)/(1 + cos x))"d"x`

Exercise 9.3 | Q 1. (v) | Page 112

Evaluate the following definite integrals:

`int_0^(pi/2) sqrt(cos theta) sin^3theta  "d"theta`

Exercise 9.3 | Q 1. (vi) | Page 112

Evaluate the following definite integrals:

`int_0^1 (1 - x^2)/(1 + x^2)^2  "d"x`

Exercise 9.3 | Q 2. (i) | Page 113

Evaluate the following integrals using properties of integration:

`int_(-5)^5 x cos(("e"^x - 1)/("e"^x + 1))  "d"x`

Exercise 9.3 | Q 2. (ii) | Page 113

Evaluate the following integrals using properties of integration:

`int_(- pi/2)^(pi/2) (x^5 + x cos x + tan^3 x + 1)  "d"x`

Exercise 9.3 | Q 2. (iii) | Page 113

Evaluate the following integrals using properties of integration:

`int_(- pi/4)^(pi/4) sin^2x  "d"x`

Exercise 9.3 | Q 2. (iv) | Page 113

Evaluate the following integrals using properties of integration:

`int_0^(2pi) x log((3 + cosx)/(3 - cosx)) "d"x`

Exercise 9.3 | Q 2. (v) | Page 113

Evaluate the following integrals using properties of integration:

`int_0^pi sin^4 x cos^3 x  "d"x`

Exercise 9.3 | Q 2. (vi) | Page 113

Evaluate the following integrals using properties of integration:

`int_0^1 |5x - 3|  "d"x`

Exercise 9.3 | Q 2. (vii) | Page 113

Evaluate the following integrals using properties of integration:

`int_0^(sin^2x) sin^-1 sqrt("t")  "dt" + int_0^(cos^2x) cos^-1 sqrt("t")  "dt"`

Exercise 9.3 | Q 2. (viii) | Page 113

Evaluate the following integrals using properties of integration:

`int_0^1 (log(1 + x))/(1 + x^2)  "d"x`

Exercise 9.3 | Q 2. (ix) | Page 113

Evaluate the following integrals using properties of integration:

`int_0^pi(xsinx)/(1 + sinx)  "'d"x`

Exercise 9.3 | Q 2. (x) | Page 113

Evaluate the following integrals using properties of integration:

`int_(pi/8)^((3pi)/8) 1/(1 + sqrt(tan x))  "d"x`

Exercise 9.3 | Q 2. (xi) | Page 113

Evaluate the following integrals using properties of integration:

`int_0^pi x[sin^2(sin x) cos^2 (cos x)] "d"x`

Exercise 9.4 [Page 115]

Tamil Nadu Board Samacheer Kalvi solutions for Class 12th Mathematics Volume 1 and 2 Answers Guide Chapter 9 Applications of IntegrationExercise 9.4 [Page 115]

Exercise 9.4 | Q 1 | Page 115

Evaluate the following:

`int_0^1 x^3"e"^(-2x)  "d"x`

Exercise 9.4 | Q 2 | Page 115

Evaluate the following:

`int_0^1 (sin(3tan^-1 x)tan^-1 x)/(1 + x^2)  "d"x`

Exercise 9.4 | Q 3 | Page 115

Evaluate the following:

`int_0^(1/sqrt(2)) ("e"^(sin^-1x) sin^-1 x)/sqrt(1 - x^2)  "d"x`

Exercise 9.4 | Q 4 | Page 115

Evaluate the following:

`int_0^(pi/2) x^2 cos 2x  "d"x`

Exercise 9.5 [Page 117]

Tamil Nadu Board Samacheer Kalvi solutions for Class 12th Mathematics Volume 1 and 2 Answers Guide Chapter 9 Applications of IntegrationExercise 9.5 [Page 117]

Exercise 9.5 | Q 1. (i) | Page 117

Evaluate the following:

`int_0^(pi/2) ("d"x)/(1 + 5cos^2x)`

Exercise 9.5 | Q 1. (ii) | Page 117

Evaluate the following:

`int_0^(pi/2) ("d"x)/(5 + 4sin^2x)`

Exercise 9.6 [Page 120]

Tamil Nadu Board Samacheer Kalvi solutions for Class 12th Mathematics Volume 1 and 2 Answers Guide Chapter 9 Applications of IntegrationExercise 9.6 [Page 120]

Exercise 9.6 | Q 1. (i) | Page 120

Evaluate the following:

`int_0^(pi/2) sin^10 x  "d"x`

Exercise 9.6 | Q 1. (ii) | Page 120

Evaluate the following:

`int_0^(pi/2) cos^7 x  "d"x`

Exercise 9.6 | Q 1. (iii) | Page 120

Evaluate the following:

`int_0^(pi/4) sin^6 2x  "d"x`

Exercise 9.6 | Q 1. (iv) | Page 120

Evaluate the following:

`int_0^(pi/6) sin^5 3x  "d"x`

Exercise 9.6 | Q 1. (v) | Page 120

Evaluate the following:

`int_0^(pi/2) sin^2x cos^4 x  "d"x`

Exercise 9.6 | Q 1. (vi) | Page 120

Evaluate the following:

`int_0^(2pi) sin^7  x/4  "d"x`

Exercise 9.6 | Q 1. (vii) | Page 120

Evaluate the following:

`int_0^(pi/2) sin^3theta cos^5theta  "d"theta`

Exercise 9.6 | Q 1. (viii) | Page 120

Evaluate the following:

`int_1^0 x^2 (1 - x)^3  "d"x`

Exercise 9.7 [Page 122]

Tamil Nadu Board Samacheer Kalvi solutions for Class 12th Mathematics Volume 1 and 2 Answers Guide Chapter 9 Applications of IntegrationExercise 9.7 [Page 122]

Exercise 9.7 | Q 1. (i) | Page 122

Evaluate the following:

`int_0^oo x^5 "e"^(-3x)  "d"x`

Exercise 9.7 | Q 1. (ii) | Page 122

Evaluate the following:

`int_0^(pi/2) ("e"^(-tanx))/(cos^6x)  "d"x`

Exercise 9.7 | Q 2 | Page 122

Evaluate the following:

If `int_0^oo "e"^(-"a"x^2) x^3  "d"x` = 32, `alpha > 0`, find `alpha`

Exercise 9.8 [Pages 134 - 135]

Tamil Nadu Board Samacheer Kalvi solutions for Class 12th Mathematics Volume 1 and 2 Answers Guide Chapter 9 Applications of IntegrationExercise 9.8 [Pages 134 - 135]

Exercise 9.8 | Q 1 | Page 134

Find the area of the region bounded by 3x – 2y + 6 = 0, x = – 3, x = 1 and x-axis

Exercise 9.8 | Q 2 | Page 134

Find the area of the region bounded by 2x – y + 1 = 0, y = – 1, y = 3 and y-axis

Exercise 9.8 | Q 3 | Page 134

Find the area of the region bounded by the curve 2 + x – x2 + y = 0, x axis, x = – 3 and x = 3

Exercise 9.8 | Q 4 | Page 134

Find the area of the region bounded by the line y = 2x + 5 and the parabola y = x2 – 2x

Exercise 9.8 | Q 5 | Page 134

Find the area of the region bounded between the curves y = sin x and y = cos x and the lines x = 0 and x = π

Exercise 9.8 | Q 6 | Page 134

Find the area of the region bounded by y = tan x, y = cot x and the lines x = 0, x = `pi/2`, y = 0

Exercise 9.8 | Q 7 | Page 134

Find the area of the region bounded by the parabola y2 = x and the line y = x – 2

Exercise 9.8 | Q 8 | Page 135

Father of a family wishes to divide his square field bounded by x = 0, x = 4, y = 4 and y = 0 along the curve y2 = 4x and x2 = 4y into three equal parts for his wife, daughter and son. Is it possible to divide? If so, find the area to be divided among them

Exercise 9.8 | Q 9 | Page 135

The curve y = (x – 2)2 + 1 has a minimum point at P. A point Q on the curve is such that the slope of PQ is 2. Find the area bounded by the curve and the chord PQ

Exercise 9.8 | Q 10 | Page 135

Find the area of the region common to the circle x2 + y2 = 16 and the parabola y2 = 6x

Exercise 9.9 [Page 139]

Tamil Nadu Board Samacheer Kalvi solutions for Class 12th Mathematics Volume 1 and 2 Answers Guide Chapter 9 Applications of IntegrationExercise 9.9 [Page 139]

Exercise 9.9 | Q 1 | Page 139

Find by integration, the volume of the solid generated by revolving about the x-axis, the region enclosed by y = 2x2, y = 0 and x = 1

Exercise 9.9 | Q 2 | Page 139

Find, by integration, the volume of the solid generated by revolving about the x axis, the region enclosed by y = e-2x, y = 0, x = 0 and x = 1

Exercise 9.9 | Q 3 | Page 139

Find, by integration, the volume of the solid generated by revolving about the y axis, the region enclosed by x2 = 1 + y and y = 3

Exercise 9.9 | Q 4 | Page 139

The region enclosed between the graphs of y = x and y = x2 is denoted by R. Find the volume generated when R is rotated through 360° about x-axis

Exercise 9.9 | Q 5 | Page 139

Find, by integration, the volume of the container which is in the shape of a right circular conical frustum as shown in the Fig 9.46

Exercise 9.9 | Q 6 | Page 139

A watermelon has an ellipsoid shape which can be obtained by revolving an ellipse with major-axis 20 cm and minor-axis 10 cm about its major-axis. Find its volume using integration

Exercise 9.10 [Pages 139 - 141]

Tamil Nadu Board Samacheer Kalvi solutions for Class 12th Mathematics Volume 1 and 2 Answers Guide Chapter 9 Applications of IntegrationExercise 9.10 [Pages 139 - 141]

MCQ

Exercise 9.10 | Q 1 | Page 139

Choose the correct alternative:

The value of `int_0^(2/3) ("d"x)/sqrt(4 - 9x^2)` is

  • `pi/6`

  • `pi/2`

  • `pi/4`

  • `pi`

Exercise 9.10 | Q 2 | Page 140

Choose the correct alternative:

The value of `int_(-1)^2 |x|  "d"x` is

  • `1/2`

  • `3/2`

  • `5/2`

  • `7/2`

Exercise 9.10 | Q 3 | Page 140

Choose the correct alternative:

For any value of n ∈ Z, `int_0^pi "e"^(cos^2x) cos^3[(2n+ 1)x]  "d"x` is

  • `pi/2`

  • `pi`

  • 0

  • 2

Exercise 9.10 | Q 4 | Page 140

Choose the correct alternative:

The value of `int_(- pi/2)^(pi/2) sin^2x cos x  "d"x` is

  • `3/2`

  • `1/2`

  • 0

  • `2/3`

Exercise 9.10 | Q 5 | Page 140

Choose the correct alternative:

The value of `int_(-4)^4 [tan^-1  ((x^2)/(x^4 + 1)) + tan^-1 ((x^4 + 1)/x^2)] "d"x` is

  • `pi`

  • `2pi`

  • `3pi`

  • `4pi`

Exercise 9.10 | Q 6 | Page 140

Choose the correct alternative:

The value of `int_(- pi/4)^(pi/4) ((2x^7 - 3x^5 + 7x^3 - x + 1)/(cos^2x)) "d"x` is

  • 4

  • 3

  • 2

  • 0

Exercise 9.10 | Q 7 | Page 140

Choose the correct alternative:

If `f(x) = int_0^x "t" cos  "t"  "dt"`, then `("d"f)/("d"x)` =

  • cos x − x sin x

  • sin x + x cos x

  • x cos x

  • x sin x

Exercise 9.10 | Q 8 | Page 140

Choose the correct alternative:

The area between y2 = 4x and its latus rectum is

  • `2/3`

  • `4/3`

  • `8/3`

  • `5/3`

Exercise 9.10 | Q 9 | Page 140

Choose the correct alternative:

The value of `int_0^1 x(1 - x)^99  "d"x` is

  • `1/11000`

  • `1/10100`

  • `1/10010`

  • `1/10001`

Exercise 9.10 | Q 10 | Page 140

Choose the correct alternative:

The value of `int_0^pi ("d"x)/(1 + 5^(cosx))` is

  • `pi/2`

  • π

  • `(3pi)/2`

Exercise 9.10 | Q 11 | Page 140

Choose the correct alternative:

If `("I'"("n" + 2))/("I'n")` = 90 then n is

  • 10

  • 5

  • 8

  • 9

Exercise 9.10 | Q 12 | Page 140

Choose the correct alternative:

The value of `int_0^(pi/6) cos^3 3x  "d"x` is

  • `2/3`

  • `2/9`

  • `1/9`

  • `1/3`

Exercise 9.10 | Q 13 | Page 141

Choose the correct alternative:

The value of  `int_10^pi sin^4x  "d"x`

  • `(3pi)/10`

  • `(3pi)/8`

  • `(3pi)/4`

  • `(3pi)/2`

Exercise 9.10 | Q 14 | Page 141

Choose the correct alternative:

The value of `int_0^oo "e"^(-3x) x^2  "d"x` is

  • `7/27`

  • `5/27`

  • `4/27`

  • `2/27`

Exercise 9.10 | Q 15 | Page 141

Choose the correct alternative:

If `int_0^"a" 1/(4 + x^2)  "dx=pi/8` then a is

  • 4

  • 1

  • 3

  • 2

Exercise 9.10 | Q 16 | Page 141

Choose the correct alternative:

The volume of solid of revolution of the region bounded by y2 = x(a – x) about the x-axis is

  • πa3

  • `(pi"a"^3)/4`

  • `(pi"a"^3)/5`

  • `(pi"a"^3)/6`

Exercise 9.10 | Q 17 | Page 141

Choose the correct alternative:

If `f(x) = int_1^x "e"^(sin u)/u  "d"u, x > 1` and `int_1^3 "e"^(sin x^2)/x  "d"x = 1/2 [f("a") - f(1)]`. then one of the possible value of a is

  • 3

  • 6

  • 9

  • 5

Exercise 9.10 | Q 18 | Page 141

Choose the correct alternative:

The value of `int_0^1 (sin^-1x)^2  "d"x` is

  • `pi^2/4 - 1`

  • `pi^2/4 + 2`

  • `pi^2/4 + 1`

  • `pi^2/4 - 2`

Exercise 9.10 | Q 19 | Page 141

Choose the correct alternative:

The value of `int_0^"a" (sqrt("a"^2 - x^2))^3  "d"x` is

  • `(pi"a"^3)/16`

  • `(3pi"a"^4)/16`

  • `(3pi"a"^2)/8`

  • `(3pi"a"^4)/8`

Exercise 9.10 | Q 20 | Page 141

Choose the correct alternative:

If `int_0^x f("t")  "dt" = x + int_x^1 "t" f("t")  "dt"`, then the value of `f(1)` is

  • `1/2`

  • 2

  • 1

  • `3/4`

Chapter 9: Applications of Integration

Exercise 9.1Exercise 9.2Exercise 9.3Exercise 9.4Exercise 9.5Exercise 9.6Exercise 9.7Exercise 9.8Exercise 9.9Exercise 9.10
Class 12th Mathematics Volume 1 and 2 Answers Guide - Shaalaa.com

Tamil Nadu Board Samacheer Kalvi solutions for Class 12th Mathematics Volume 1 and 2 Answers Guide chapter 9 - Applications of Integration

Tamil Nadu Board Samacheer Kalvi solutions for Class 12th Mathematics Volume 1 and 2 Answers Guide chapter 9 (Applications of Integration) include all questions with solution and detail explanation. This will clear students doubts about any question and improve application skills while preparing for board exams. The detailed, step-by-step solutions will help you understand the concepts better and clear your confusions, if any. Shaalaa.com has the Tamil Nadu Board of Secondary Education Class 12th Mathematics Volume 1 and 2 Answers Guide solutions in a manner that help students grasp basic concepts better and faster.

Further, we at Shaalaa.com provide such solutions so that students can prepare for written exams. Tamil Nadu Board Samacheer Kalvi textbook solutions can be a core help for self-study and acts as a perfect self-help guidance for students.

Concepts covered in Class 12th Mathematics Volume 1 and 2 Answers Guide chapter 9 Applications of Integration are Applications of Integrations, Definite Integral as the Limit of a Sum, Fundamental Theorems of Integral Calculus and Their Applications, Bernoulli’s Formula, Improper Integrals, Reduction Formulae, Gamma Integral, Evaluation of a Bounded Plane Area by Integration, Volume of a Solid Obtained by Revolving Area About an Axis.

Using Tamil Nadu Board Samacheer Kalvi Class 12th solutions Applications of Integration exercise by students are an easy way to prepare for the exams, as they involve solutions arranged chapter-wise also page wise. The questions involved in Tamil Nadu Board Samacheer Kalvi Solutions are important questions that can be asked in the final exam. Maximum students of Tamil Nadu Board of Secondary Education Class 12th prefer Tamil Nadu Board Samacheer Kalvi Textbook Solutions to score more in exam.

Get the free view of chapter 9 Applications of Integration Class 12th extra questions for Class 12th Mathematics Volume 1 and 2 Answers Guide and can use Shaalaa.com to keep it handy for your exam preparation

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