#### Chapters

Chapter 2: Complex Numbers

Chapter 3: Theory of Equations

Chapter 4: Inverse Trigonometric Functions

Chapter 5: Two Dimensional Analytical Geometry-II

Chapter 6: Applications of Vector Algebra

Chapter 7: Applications of Differential Calculus

Chapter 8: Differentials and Partial Derivatives

Chapter 9: Applications of Integration

Chapter 10: Ordinary Differential Equations

Chapter 11: Probability Distributions

Chapter 12: Discrete Mathematics

## 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 Exercise 9.1 [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}

Find an approximate value of `int_1^1.5` x^{2}dx by applying the right–end rule with the partition {1.1, 1.2, 1.3, 1.4, 1.5}

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}

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

Evaluate the following integrals as the limits of sum:

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

Evaluate the following integrals as the limits of sum:

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

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

Evaluate the following definite integrals:

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

Evaluate the following definite integrals:

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

Evaluate the following definite integrals:

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

Evaluate the following definite integrals:

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

Evaluate the following definite integrals:

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

Evaluate the following definite integrals:

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

Evaluate the following integrals using properties of integration:

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

Evaluate the following integrals using properties of integration:

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

Evaluate the following integrals using properties of integration:

`int_(- pi/4)^(pi/4) sin^2x "d"x`

Evaluate the following integrals using properties of integration:

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

Evaluate the following integrals using properties of integration:

`int_0^pi sin^4 x cos^3 x "d"x`

Evaluate the following integrals using properties of integration:

`int_0^1 |5x - 3| "d"x`

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"`

Evaluate the following integrals using properties of integration:

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

Evaluate the following integrals using properties of integration:

`int_0^pi(xsinx)/(1 + sinx) "'d"x`

Evaluate the following integrals using properties of integration:

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

Evaluate the following integrals using properties of integration:

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

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

Evaluate the following:

`int_0^1 x^3"e"^(-2x) "d"x`

Evaluate the following:

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

Evaluate the following:

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

Evaluate the following:

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

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

Evaluate the following:

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

Evaluate the following:

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

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

Evaluate the following:

`int_0^(pi/2) sin^10 x "d"x`

Evaluate the following:

`int_0^(pi/2) cos^7 x "d"x`

Evaluate the following:

`int_0^(pi/4) sin^6 2x "d"x`

Evaluate the following:

`int_0^(pi/6) sin^5 3x "d"x`

Evaluate the following:

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

Evaluate the following:

`int_0^(2pi) sin^7 x/4 "d"x`

Evaluate the following:

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

Evaluate the following:

`int_1^0 x^2 (1 - x)^3 "d"x`

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

Evaluate the following:

`int_0^oo x^5 "e"^(-3x) "d"x`

Evaluate the following:

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

Evaluate the following:

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

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

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

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

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

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

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

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

Find the area of the region bounded by the parabola y^{2} = x and the line y = x – 2

Father of a family wishes to divide his square field bounded by x = 0, x = 4, y = 4 and y = 0 along the curve y^{2} = 4x and x^{2} = 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

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

Find the area of the region common to the circle x^{2} + y^{2} = 16 and the parabola y^{2} = 6x

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

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

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

Find, by integration, the volume of the solid generated by revolving about the y axis, the region enclosed by x^{2} = 1 + y and y = 3

The region enclosed between the graphs of y = x and y = x^{2} is denoted by R. Find the volume generated when R is rotated through 360° about x-axis

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

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

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

#### MCQ

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`

Choose the correct alternative:

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

`1/2`

`3/2`

`5/2`

`7/2`

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

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`

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`

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

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

Choose the correct alternative:

The area between y^{2} = 4x and its latus rectum is

`2/3`

`4/3`

`8/3`

`5/3`

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`

Choose the correct alternative:

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

`pi/2`

π

`(3pi)/2`

2π

Choose the correct alternative:

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

10

5

8

9

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`

Choose the correct alternative:

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

`(3pi)/10`

`(3pi)/8`

`(3pi)/4`

`(3pi)/2`

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`

Choose the correct alternative:

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

4

1

3

2

Choose the correct alternative:

The volume of solid of revolution of the region bounded by y^{2} = x(a – x) about the x-axis is

πa

^{3}`(pi"a"^3)/4`

`(pi"a"^3)/5`

`(pi"a"^3)/6`

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

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`

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`

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

## 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.

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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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