### myCBSEguide App

Download the app to get CBSE Sample Papers 2023-24, NCERT Solutions (Revised), Most Important Questions, Previous Year Question Bank, Mock Tests, and Detailed Notes.

Install NowNCERT Solutions class 12 Maths Miscellaneous Class 12 Maths book solutions are available in PDF format for free download. These ncert book chapter wise questions and answers are very helpful for CBSE board exam. CBSE recommends NCERT books and most of the questions in CBSE exam are asked from NCERT text books. Class 12 Maths chapter wise NCERT solution for Maths part 1 and Maths part 2 for all the chapters can be downloaded from our website and myCBSEguide mobile app for free.

**Download NCERT solutions for Applications of Integrals as PDF.**

## NCERT Solutions class 12 Maths Applications of Integrals

**1. Find the area under the given curves and given lines:**

**(i)**** and ****axis.**

**(ii)**** and ****axis.**

**Ans. (i)**Equation of the curve (parabola) is

……..(i)

Required area bounded by curve (i), vertical line and axis

=

=

=

= sq. units

** (ii)**Equation of the curve

…..(i)

It is clear that curve (i) passes through the origin because from (i)

Table of values for curve for and (given)

1 | 2 | 3 | 4 | 5 | |

1 |

Required shaded area between the curve , vertical lines and axis

= =

= =

=

= 624.8 sq. units

### NCERT Solutions class 12 Maths Miscellaneous

**2. Find the area between the curves **** and **** **

**Ans. **Equation of one curve (straight line) is …..(i)

Equation of second curve (parabola) is …..(ii)

Solving eq. (i) and (ii), we get or and or

Points of intersection of line (i) and parabola (ii) are O (0, 0) and A (1, 1).

Now Area of triangle OAM

= Area bounded by line (i) and axis

= = =

= sq. units

Also Area OBAM = Area bounded by parabola (ii) and axis

= = =

= sq. units

Required area OBA between line (i) and parabola (ii)

= Area of triangle OAM – Area of OBAM

= = sq. units

### NCERT Solutions class 12 Maths Miscellaneous

**3. Find the area of the region lying in the first quadrant and bounded by **** and **** **

**Ans. **Equation of the curve (parabola) is

……….(i)

……….(ii)

Here required shaded area of the region lying in first quadrant bounded by parabola (i),

and the horizontal lines and is

= =

=

=

=

= = sq. units

### NCERT Solutions class 12 Maths Miscellaneous

**4. Sketch the graph of **** and evaluate **** **

**Ans. **Equation of the given curve is ……….(i)

0 for all real

Graph of curve is only above the axis i.e., in first and second quadrant only.

=

If 0

….(ii)

And

=

If 0

…….(iii)

Table of values for for

0 | ||||

0 | 1 | 2 | 3 |

Table of values for for

0 | 1 | 2 | 3 |

Now,

= +

= +

=

=

=

= sq. units

### NCERT Solutions class 12 Maths Miscellaneous

**5. Find the area bounded by the curve **** between **** and **** **

**Ans. **Equation of the curve is ……….(i)

for i.e., graph is in first and second quadrant.

And for i.e., graph is in third and fourth quadrant.

If tangent is parallel to axis, then

Table of values for curve between and

0 | |||||

0 | 1 | 0 | 0 |

Now Required shaded area = Area OAB + Area BCD

=

=

=

=

=

= 2 + 2 = 4 sq. units

### NCERT Solutions class 12 Maths Miscellaneous

**6. Find the area enclosed by the parabola **** and the line **** **

**Ans. **Equation of parabola is ……..(i)

The area enclosed between the parabola and line is represented by shaded area OADO.

Here Points of intersection of curve (i) and line are O (0, 0) and A.

Now Area ODAM = Area of parabola and axis

=

=

=

=

= ……….(ii)

Again Area of = Area between line and axis

= =

=

= = ……….(ii)

Requires shaded area = Area ODAM – Area of

=

=

=

### NCERT Solutions class 12 Maths Miscellaneous

**7. Find the area enclosed by the parabola **** and the line **** **

**Ans. **Equation of the parabola is

……….(i)

Equation of the line is ……(ii)

In the graph, points of intersection are B (4, 12) and C.

Now, Area ABCD =

=

=

= 45 sq. units

Again, Area CDO + Area OAB =

=

= = 18 sq. units

Required area = Area ABCD – (Area CDO + Area OAB)

= 45 – 18 = 27 sq. units

### NCERT Solutions class 12 Maths Miscellaneous

**8. Find the area of the smaller region bounded by the ellipse **** and the line **** **

**Ans. **Equation of the ellipse is

………(i)

Here intersection of ellipse (i) with axis are

A (3, 0) and A’ and intersection of ellipse (i) with axis are B (0, 2) and B’.

Also, the points of intersections of ellipse (i) and line are A (3, 0) and B (0, 2).

Area OADB = Area between ellipse (i) (arc AB of it) and axis

=

=

=

=

=

= = sq. units……….(ii)

Again Area of triangle OAB = Area bounded by line AB and axis

=

=

=

= sq. units ….(iii)

Now Required shaded area = Area OADB – Area OAB

=

= sq. units

### NCERT Solutions class 12 Maths Miscellaneous

**9. Find the area of the smaller region bounded by the ellipse **** and the line **** **

**Ans. **Equation of ellipse is ……..(i)

Area between arc AB of the ellipse and axis

=

=

=

=

= ………..(ii)

Also Area between chord AB and axis

=

=

=

=

=

Now Required area

= Area between arc AB of the ellipse and axis – Area between chord AB and axis

= sq. units

**10.Find the area of the region enclosed by the parabola **** the line **** and **** axis.**

**Ans. **Equation of parabola is …………(i)

Equation of line is …………(ii)

Here the two points of intersections of parabola (i) and line (ii) are A and B (2, 4).

Area ALODBM = Area bounded by parabola (i) and axis

= =

=

= = = 3 sq. units

AlsoArea of trapezium ALMB = Area bounded by line (ii) and axis

= =

=

=

= sq. units

Now Required area = Area of trapezium ALMB – Area ALODBM

= sq. units

**11.Using the method of integration, find the area enclosed by the curve **** **

**Ans. **Equation of the curve (graph) is

……….(i)

The area bounded by the curve (i) is represented by the shaded region ABCD.

The curve intersects the axes at points A (1, 0), B (0, 1), C and D

It is observed clearly that given curve is symmetrical about axis and axis.

Area bounded by the curve

= Area of square ABCD

= 4 x OAB

=

=

=

= = 2 sq. units

**12.Find the area bounded by the curves ****.**

**Ans. **The area bounded by the curves is represented by the shaded region.

It is clearly observed that the required area is symmetrical about axis.

Required area

= Area between parabola and axis between limits and

= =

= ………..(i)

And Area of ray and axis,

= = = ………..(ii)

Required shaded area in first quadrant

= Area between ray for and axis – Area between parabola and axis in first quadrant

= Area given by eq. (ii) – Area given by eq. (i) = sq. units

**13.Using the method of integration, find the area of the triangle whose vertices are A (2, 0), B (4, 5) and C (6, 3).**

**Ans. **Vertices of the given triangle are A (2, 0), B (4, 5) and C (6, 3).

Equation of side AB is

=

Equation of side BC is

=

Equation of side AC is

=

Now, Required shaded area = Area + Area of trapezium BLMC – Area

=

=

=

=

=

= 5 + 8 – 6 = 7 sq. units

**14.Using the method of integration, find the area of the region bounded by the lines: **** **** and **** **

**Ans. **Equation of one line is ,

Equation of second line is

And Equation of third line is

Here, vertices of triangle ABC are A (2, 0), B (4, 3) and C (1, 2).

Now, Required area of triangle

= Area of trapezium CLMB – Area Area

=

=

=

=

=

= sq. units

**15.Find the area of the region ****.**

**Ans. **Equation of parabola is ……(i)

And equation of circle is ……(ii)

Here, the two points of intersection of parabola (i) and circle (ii) are A and B

Required shaded area OADBO (Area of the circle which is interior to the parabola)

= 2 x Area OADO = 2 [Area OAC + Area CAD]

=

=

=

=

= sq. units

**16.Choose the correct answer:**

**Area bounded by the curve the axis and the ordinate and is:**

**(A) **

**(B) **

**(C) **

**(D) **

**Ans. **Equation of the curve is **Y**

To find: Area OBN ( for ) and Area OAM ( for )

Required area = Area OBN + Area OAM

=

=

=

= = sq. units

Therefore, option (D) is correct,

**17.Choose the correct answer:**

**The area bounded by the curve axis and the ordinates and is given by:**

**(A) 0**

**(B) **

**(C) **

**(D) **

**Ans. **Equation of the curve is

if ……….(i)

And if ………..(ii)

Required area = Area ONBO + Area OAMO

=

=

= sq. units

Therefore, option (C) is correct.

**18.Choose the correct answer:**

**The area of the circle exterior to the parabola **

**(A) **

**(B) **

**(C) **

**(D) **

**Ans. **Equation of the circle is …..(i)

This circle is symmetrical about axis and axis.

Here two points of intersection are B and B’

Required area = Area of circle – Area of circle interior to the parabola

= Area OBAB’O

= 2 x Area OBACO

= 2[Area OBCO + Area BACB]

=

=

=

=

=

=

=

=

=

=

=

= sq. units

**19.Choose the correct answer:**

**The area bounded by the axis, and when is:**

**(A) **

**(B) **

**(C) **

**(D) **

**Ans. **Here both graphs intersect at the point B.

Required shaded area = Area OABC – Area OBC

= Area OABC – (Area OBM + Area BCM)

=

=

=

=

=

= = sq. units

Therefore, option (B) is correct.

## NCERT Solutions class 12 Maths Miscellaneous

NCERT Solutions Class 12 Maths PDF (Download) Free from myCBSEguide app and myCBSEguide website. Ncert solution class 12 Maths includes text book solutions from both part 1 and part 2. NCERT Solutions for CBSE Class 12 Maths have total 20 chapters. 12 Maths NCERT Solutions in PDF for free Download on our website. Ncert Maths class 12 solutions PDF and Maths ncert class 12 PDF solutions with latest modifications and as per the latest CBSE syllabus are only available in myCBSEguide

## CBSE App for Class 12

To download NCERT Solutions for class 12 Physics, Chemistry, Biology, History, Political Science, Economics, Geography, Computer Science, Home Science, Accountancy, Business Studies and Home Science; do check myCBSEguide app or website. myCBSEguide provides sample papers with solution, test papers for chapter-wise practice, NCERT solutions, NCERT Exemplar solutions, quick revision notes for ready reference, CBSE guess papers and CBSE important question papers. Sample Paper all are made available through **the best app for CBSE students** and myCBSEguide website

Test Generator

Create question paper PDF and online tests with your own name & logo in minutes.

Create NowmyCBSEguide

Question Bank, Mock Tests, Exam Papers, NCERT Solutions, Sample Papers, Notes

Install Now