Tamil Nadu Board of Secondary EducationHSC Science Class 12th

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

## 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
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

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

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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.

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