Draw a rough sketch of the region ( x , y ) : y 2 ≤ 3 x , 3 x 2 + 3 y 2 ≤ 16 and find the area enclosed by the region, using the method of integration.

Given that the boundaries of the area to be found are R = ( x , y ) : y 2 ≤ 3 x , 3 x 2 + 3 y 2 ≤ 16 This can be written as R 1 = ( x , y ) : y 2 ≤ 3 x R 2 = ( x , y ) : 3 x 2 + 3 y 2 ≤ 16 Area required is = R 1 ∩ R 2 y 2 = 3 x . . . … . . ( 1 ) Symmetric about the x - axis, as it has the even power of y 3 x 2 + 3 y 2 = 16 . . . … . . ( 2 ) Now to find the point of intersection of ( 1 ) and ( 2 ) , substitute y 2 = 3 x in ( 2 ) 3 x 2 + 3 ( 3 x ) = 16 3 x 2 + 9 x - 16 = 0 x = - 9 ± 9 2 - 4 ( 3 ) ( - 16 ) 2 ( 3 ) = - 9 ± 81 + 192 6 = - 9 ± 273 6 as y cannot be imaginary, we reject the negative value of x S o x = - 9 + 273 6 So the two points, A and B are the points where ( 1 ) and ( 2 ) meet. The line AB meets the x - axis at D = - 9 + 273 6 , 0 Substitute y = 0 in ( 2 ) , 3 x 2 + 0 = 16 x = ± 4 3 So the circle intersects the x - axis at C = 4 3 , 0 and E = - 4 3 , 0 As x and y have even powers for the circle, they will be symmetrical about the x - axis and y - axis. Consider the circle, 3 x 2 + 3 y 2 = 16 , can be re-written as x 2 + y 2 = 16 3 y = 16 3 - x 2 . . . . . . . . . ( 3 ) Consider the parabola, y 2 = 3 x , can be re-written as y = 3 x - … ( 4 ) Now, the area to be found will be the area is Area of the required region = Area of O A C B O . Area of O A B C O = Area of O C A O + Area of O C B O [area of O C B O = area of O C A O as the circle is symmetrical about the y - axis] Area of O A C B O = 2 × Area of O C A O - ( 5 ) Area of O C A O = Area of O A D O + Area of D A C D Area of OCAO = ∫ 0 - 9 + 273 6 3 x dx + ∫ - 9 + 273 6 4 3 16 3 - x 2 dx = 3 x 3 2 3 2 0 - 9 + 273 6 + x 16 3 - x 2 2 + 16 3 2 sin - 1 x 4 3 - 9 + 273 6 4 3 Let - 9 + 273 6 = a = 2 3 3 a 3 2 - 0 3 2 + 4 3 16 3 - 4 3 2 2 + 16 3 2 sin - 1 4 3 4 3 - a 16 3 - ( a ) 2 2 + 16 3 2 sin - 1 a 4 3 = 2 3 3 a 3 2 + 0 + 16 6 sin - 1 ( 1 ) - a 16 3 - ( a ) 2 2 + 16 6 sin - 1 3 a 4 = 2 3 3 a 3 2 + 16 6 π 2 - a 16 3 - ( a ) 2 2 + 16 6 sin - 1 3 a 4 = 2 3 3 a 3 2 + 4 π 3 - a 16 3 - ( a ) 2 2 - 8 3 sin - 1 3 a 4 The Area of O C A O = 2 3 3 a 3 2 + 4 π 3 - a 16 3 - ( a ) 2 2 - 8 3 sin - 1 3 a 4 sq . units , where a = - 9 + 273 6 Now substituting the area of O C A O in equation ( 5 ) Area of O A C B O = 2 × Area of O C A O = 2 2 3 3 a 3 2 + 4 π 3 - a 16 3 - ( a ) 2 2 + 8 3 sin - 1 3 a 4 Area of the required region = 4 3 3 a 3 2 - 8 π 3 - a 16 3 - ( a ) 2 - 16 3 sin - 1 3 a 4 sq . units where a = - 9 + 273 6

E M B I B E

SENIOR SECONDARY SCHOOL MATHEMATICS FOR CLASS 12>Chapter 17 - Area of Bounded Regions>EXERCISE>Q 13

HARD

12th CBSE

IMPORTANT

Earn 100

Using integration, find the area of the region bounded between the line $x = 2$ and the parabola $y^{2} = 8 x$ .

Important Questions on Area of Bounded Regions

EASY

12th CBSE

IMPORTANT

SENIOR SECONDARY SCHOOL MATHEMATICS FOR CLASS 12>Chapter 17 - Area of Bounded Regions>EXERCISE>Q 14

Using integration, find the area of region bounded by the line $y - 1 = x$ , the $x$ -axis, and the ordinates $x = - 2$ and $x = 3$ .

EASY

12th CBSE

IMPORTANT

SENIOR SECONDARY SCHOOL MATHEMATICS FOR CLASS 12>Chapter 17 - Area of Bounded Regions>EXERCISE>Q 15

Sketch the region lying in the first quadrant and bounded by $y = 4 x^{2}, x = 0, y = 2$ and $y = 4$ . Find the area of the region using integration.

EASY

12th CBSE

IMPORTANT

SENIOR SECONDARY SCHOOL MATHEMATICS FOR CLASS 12>Chapter 17 - Area of Bounded Regions>EXERCISE>Q 16

Sketch the region lying in the first quadrant and bounded by $y = 9 x^{2}, x = 0, y = 1$ and $y = 4$ . Find the area of the region, using integration.

HARD

12th CBSE

IMPORTANT

SENIOR SECONDARY SCHOOL MATHEMATICS FOR CLASS 12>Chapter 17 - Area of Bounded Regions>EXERCISE>Q 17

Find the area of the region enclosed between the circles $x^{2} + y^{2} = 1$ and ${(x - 1)}^{2} + y^{2} = 1$

HARD

12th CBSE

IMPORTANT

SENIOR SECONDARY SCHOOL MATHEMATICS FOR CLASS 12>Chapter 17 - Area of Bounded Regions>EXERCISE>Q 18

Sketch the region common to the circle $x^{2} + y^{2} = 16$ and the parabola $x^{2} = 6 y$ . Also, find the area of the region, using integration.

HARD

12th CBSE

IMPORTANT

SENIOR SECONDARY SCHOOL MATHEMATICS FOR CLASS 12>Chapter 17 - Area of Bounded Regions>EXERCISE>Q 19

Sketch the region common to the circle $x^{2} + y^{2} = 25$ and the parabola $y^{2} = 8 x$ . Also, find the area of the region, using integration.

HARD

12th CBSE

IMPORTANT

SENIOR SECONDARY SCHOOL MATHEMATICS FOR CLASS 12>Chapter 17 - Area of Bounded Regions>EXERCISE>Q 20

Draw a rough sketch of the region $\{(x, y) : y^{2} \leq 3 x, 3 x^{2} + 3 y^{2} \leq 16\}$ and find the area enclosed by the region, using the method of integration.