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Exercise 8.2 - Chapter 8 Applications Of Integrals class 12 ncert solutions Maths - SaraNextGen [2024]


Question 1:

Find the area of the circle 4x2 + 4y2 = 9 which is interior to the parabola x2 = 4y

Answer:

The required area is represented by the shaded area OBCDO.

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/237/8006/Chapter%208_html_581266d91.jpg

Solving the given equation of circle, 4x2 + 4y2 = 9, and parabola, x2 = 4y, we obtain the point of intersection ashttps://img-nm.mnimgs.com/img/study_content/curr/1/12/15/237/8006/Chapter%208_html_2ce90f4b.gif .

It can be observed that the required area is symmetrical about y-axis.

∴ Area OBCDO = 2 × Area OBCO

We draw BM perpendicular to OA.

Therefore, the coordinates of M arehttps://img-nm.mnimgs.com/img/study_content/curr/1/12/15/237/8006/Chapter%208_html_m119f0f47.gif .

Therefore, Area OBCO = Area OMBCO – Area OMBO

=∫02(9-4×2)4dx-∫02x24dx

=∫02322-x2dx-14∫02x2dx

=x2322-x2+98sin-12×302-14×3302

=24+98sin-1223-11223

=122+98sin-1223-132

=162+98sin-1223

=1226+94sin-1223Therefore, the required area OBCDO is https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/237/8006/Chapter%208_html_26d667a8.gif units

Question 2:

Find the area bounded by curves (x – 1)2 + y2 = 1 and x2 + y 2 = 1

Answer:

The area bounded by the curves, (x – 1)2 + y2 = 1 and x2 + y 2 = 1, is represented by the shaded area as

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/237/8008/Chapter%208_html_e88a1021.jpg

On solving the equations, (x – 1)2 + y2 = 1 and x2 + y 2 = 1, we obtain the point of intersection as Ahttps://img-nm.mnimgs.com/img/study_content/curr/1/12/15/237/8008/Chapter%208_html_m5e070441.gif and Bhttps://img-nm.mnimgs.com/img/study_content/curr/1/12/15/237/8008/Chapter%208_html_m180f3cd8.gif .

It can be observed that the required area is symmetrical about x-axis.

∴ Area OBCAO = 2 × Area OCAO

We join AB, which intersects OC at M, such that AM is perpendicular to OC.

The coordinates of M are https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/237/8008/Chapter%208_html_m2b45e065.gif .

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/237/8008/Chapter%208_html_3b2645a6.gif

Therefore, required area OBCAO = https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/237/8008/Chapter%208_html_772d0eed.gif units

Question 3:

Find the area of the region bounded by the curves y = x+ 2, xx = 0 and x = 3

Answer:

The area bounded by the curves, y = x+ 2, xx = 0, and x = 3, is represented by the shaded area OCBAO as

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/237/8011/Chapter%208_html_m4200185a1.jpg

Then, Area OCBAO = Area ODBAO – Area ODCO

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/237/8011/Chapter%208_html_2cb3d0d8.gif

Question 4:

Using integration finds the area of the region bounded by the triangle whose vertices are (–1, 0), (1, 3) and (3, 2).

Answer:

BL and CM are drawn perpendicular to x-axis.

It can be observed in the following figure that,

Area (ΔACB) = Area (ALBA) + Area (BLMCB) – Area (AMCA) … (1)

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/237/8013/Chapter%208_html_37356ffe1.jpg

Equation of line segment AB is

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/237/8013/Chapter%208_html_m2c6964c4.gif

Equation of line segment BC is

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/237/8013/Chapter%208_html_6b3d802f.gif

Equation of line segment AC is

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/237/8013/Chapter%208_html_m6405a416.gif

Therefore, from equation (1), we obtain

Area (ΔABC) = (3 + 5 – 4) = 4 units

Question 5:

Using integration find the area of the triangular region whose sides have the equations y = 2x +1, y = 3x + 1 and = 4.

Answer:

The equations of sides of the triangle are y = 2x +1, y = 3x + 1, and = 4.

On solving these equations, we obtain the vertices of triangle as A(0, 1), B(4, 13), and C (4, 9).

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/237/8015/Chapter%208_html_641bfb961.jpg

It can be observed that,

Area (ΔACB) = Area (OLBAO) –Area (OLCAO)

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/237/8015/Chapter%208_html_m147728e.gif

Question 6:

Smaller area enclosed by the circle x2 + y2 = 4 and the line x + y = 2 is

A. 2 (π – 2)

B. π – 2

C. 2π – 1

D. 2 (π + 2)

Answer:

The smaller area enclosed by the circle, x2 + y2 = 4, and the line, x + y = 2, is represented by the shaded area ACBA as

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/237/8018/Chapter%208_html_746385f1.jpg

It can be observed that,

Area ACBA = Area OACBO – Area (ΔOAB)

https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/237/8018/Chapter%208_html_62cf651a.gif

Thus, the correct Answer is B.

Question 7:

Area lying between the curve y2 = 4x and y = 2x is

A. https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/237/8020/Chapter%208_html_m64d8549d.gif

B. https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/237/8020/Chapter%208_html_18a817bc.gif

C. https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/237/8020/Chapter%208_html_m3c88992d.gif

D. https://img-nm.mnimgs.com/img/study_content/curr/1/12/15/237/8020/Chapter%208_html_m2111158f.gif

Answer:

The area lying between the curve, y2 = 4x and y = 2x, is represented by the shaded area OBAO as

https://img-nm.mnimgs.com/img/study_content/editlive_ncert/33/2013_06_26_16_36_26/Grade%2012.png

The points of intersection of these curves are O (0, 0) and A (1, 2).

We draw AC perpendicular to x-axis such that the coordinates of C are (1, 0).

∴ Area OBAO = Area (OCABO) – Area (ΔOCA)

https://img-nm.mnimgs.com/img/study_content/editlive_ncert/33/2013_06_26_16_36_26/Chapter8_html_83ea2b6_8581701991481190433.png https://img-nm.mnimgs.com/img/study_content/editlive_ncert/33/2013_06_26_16_36_26/Chapter8_html_83ea2b6_1096600754141077432.png  https://img-nm.mnimgs.com/img/study_content/editlive_ncert/33/2013_06_26_16_36_26/Chapter8_html_83ea2b6_1572300423894571416.png https://img-nm.mnimgs.com/img/study_content/editlive_ncert/33/2013_06_26_16_36_26/Chapter8_html_83ea2b6_1771239675617594551.png

https://img-nm.mnimgs.com/img/study_content/editlive_ncert/33/2013_06_26_16_36_26/mathmlequation3582276327706390485.png

https://img-nm.mnimgs.com/img/study_content/editlive_ncert/33/2013_06_26_16_36_26/mathmlequation7711101480489996299.png  square units

 

Thus, the correct Answer is B.

Also Read : Miscellaneous-Exercise-Chapter-8-Applications-Of-Integrals-class-12-ncert-solutions-Maths

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