Find the area of the region bounded by y = x and y = x

We have to find area of region bounded by y 2 = x . . . 1 and y = x . . . 2 Equation 1 represents parabola with vertex 0 , 0 and axis as x -axis and equation 2 represents a line passing through origin and intersecting parabola at 0 , 0 and 1 , 1 . A rough sketch of curves is as below:- Shaded region represents the required area, We slice it in rectangle with Width = △ x length = y 1 - y 2 Area of rectangle = y 1 - y 2 △ x The approximation triangle can slide from x = 0 to x = 1 . Required area = region A Q B P A = ∫ 0 1 y 1 - y 2 d x = ∫ 0 1 x - x d x = 2 3 x x - x 2 2 0 1 = 2 3 - 1 2 2 - 0 = 2 3 - 1 2 = 1 6 square units.

If the area bounded by the parabola y 2 = 4 a x and the line y = m x is a 2 12 sq. units, then using integration, find the value of m .

The parabola y 2 = 4 a x opens towards the positive x - axis and its focus is ( a , 0 ) . The line y = m x passes through the origin ( 0 , 0 ) . Solving y 2 = 4 a x and y = m x , we get m 2 x 2 = 4 a x ⇒ m 2 x 2 - 4 a x = 0 ⇒ x m 2 x - 4 a = 0 ⇒ x = 0 or x = 4 a m 2 So, the points of intersection of the given parabola and line are O ( 0 , 0 ) and A 4 a m 2 , 4 a m ∴ Area of parabola lying in first quadrant = ∫ 0 4 a m 2 y p a r a b o l a d x = ∫ 0 4 a m 2 4 a x d x = 4 a 3 4 a m 2 3 2 - 0 = 4 a 3 × 8 a a m 3 = 32 a 2 3 m 3 Area of triangle = 1 2 × 4 a m 2 × 4 a m = 8 a 2 m 3 Area bounded by the given parabola and triangle = 32 a 2 3 m 3 - 8 a 2 m 3 = 8 a 2 3 m 3 square units But, Area bounded by the given parabola and triangle = a 2 12 sq. units (Given) ∴ 8 a 2 3 m 3 = a 2 12 ⇒ m 3 = 32 ⇒ m = 32 3 m = 2 5 3 Thus, the value of m is 2 5 3

E M B I B E

MATHEMATICS CLASS XII VOLUME-2>Chapter 2 - Areas of Bounded Regions>EXERCISE

R D Sharma Solutions for Chapter: Areas of Bounded Regions, Exercise 3: EXERCISE

Author:R D Sharma

R D Sharma Mathematics Solutions for Exercise - R D Sharma Solutions for Chapter: Areas of Bounded Regions, Exercise 3: EXERCISE

Attempt the free practice questions on Chapter 2: Areas of Bounded Regions, Exercise 3: EXERCISE with hints and solutions to strengthen your understanding. MATHEMATICS CLASS XII VOLUME-2 solutions are prepared by Experienced Embibe Experts.

Questions from R D Sharma Solutions for Chapter: Areas of Bounded Regions, Exercise 3: EXERCISE with Hints & Solutions

MEDIUM

12th CBSE

IMPORTANT

MATHEMATICS CLASS XII VOLUME-2>Chapter 2 - Areas of Bounded Regions>EXERCISE>Q 2

Find the area of the region common to the parabolas $4 y^{2} = 9 x$ and $3 x^{2} = 16 y$ .

HARD

12th CBSE

IMPORTANT

MATHEMATICS CLASS XII VOLUME-2>Chapter 2 - Areas of Bounded Regions>EXERCISE>Q 3

Find the area of the region bounded by $y = \sqrt{x}$ and $y = x$

HARD

12th CBSE

IMPORTANT

MATHEMATICS CLASS XII VOLUME-2>Chapter 2 - Areas of Bounded Regions>EXERCISE>Q 4

Find the area bounded by the curve $y = 4 - x^{2}$ and the lines $y = 0, y = 3$ .

HARD

12th CBSE

IMPORTANT

MATHEMATICS CLASS XII VOLUME-2>Chapter 2 - Areas of Bounded Regions>EXERCISE>Q 7

Using integration, find the area of the region bounded by the triangle $A B C$ whose vertices $A, B, C$ are $(- 1, 1), (0, 5)$ and $(3, 2)$ respectively.

HARD

12th CBSE

IMPORTANT

MATHEMATICS CLASS XII VOLUME-2>Chapter 2 - Areas of Bounded Regions>EXERCISE>Q 20

Find the area, lying above x-axis and included between the circle $x^{2} + y^{2} = 8 x$ and the parabola $y^{2} = 4 x$ .

MEDIUM

12th CBSE

IMPORTANT

MATHEMATICS CLASS XII VOLUME-2>Chapter 2 - Areas of Bounded Regions>EXERCISE>Q 49

In what ratio does the $x$ - axis divide the area of the region bounded by the parabolas $y = 4 x - x^{2}$ and $y = x^{2} - x ?$

MEDIUM

12th CBSE

IMPORTANT

MATHEMATICS CLASS XII VOLUME-2>Chapter 2 - Areas of Bounded Regions>EXERCISE>Q 50

Find the area of the figure (in $square units$ ) bounded by the curves $y = | x - 1 |$ and $y = 3 - | x | .$

HARD

12th CBSE

IMPORTANT

MATHEMATICS CLASS XII VOLUME-2>Chapter 2 - Areas of Bounded Regions>EXERCISE>Q 51

If the area bounded by the parabola $y^{2} = 4 a x$ and the line $y = m x$ is $\frac{a^{2}}{12}$ sq. units, then using integration, find the value of $m$ .

Learn and Prepare for any exam you want !

|

|

|

|

|

|

|

|