Find the area of the region bounded by x 2 = 4y, y = 2, y = 4 and the y-axis in the first quadrant.

Let, A B represent line y = 2 C D represent line y = 4 & C B O A D represent the curve x 2 = 4 y . We have to find the area of B C F E . Area of B C F E = 2 ∫ 2 4 x d y So, we need to calculate ∫ 2 4 x d y . We know that x 2 = 4 y Taking square root on both sides, ⇒ x = ± 2 y Since B C F E is in 1 s t Quadrant. ⇒ x = 2 y Area of B C F E = ∫ 2 4 x d y = ∫ 2 4 2 y d y = 2 ∫ 2 4 y 1 2 d y As we know, ∫ x n d x = x n + 1 n + 1 + c Area of B C F E = 2 y 1 2 + 1 1 2 + 1 2 4 = 2 y 3 2 3 2 2 4 = 4 3 y 3 2 2 4 = 4 3 4 3 2 - 2 3 2 = 4 3 4 1 2 3 - 2 3 1 2 = 4 3 2 3 - 8 1 2 = 4 3 8 - 2 2 = 8 3 4 - 2

Find the area of the region bounded by the curve ay 2 = x 3 , the y-axis and the lines y= a and y = 2a.

Let, A B represent line y = a C D represent line y = 2 a & C B O represent the curve a y 2 = x 3 . We have to find area of A B C D . Area of A B C D = ∫ a 2 a x d y So, we need to calculate ∫ a 2 a x d y . We know that a y 2 = x 3 . Taking cube root on both sides, ⇒ a 1 3 y 2 3 = x Area of A B C D = ∫ a 2 a x d y = ∫ a 2 a a 1 3 y 2 3 d y = a 1 3 ∫ a 2 a y 2 3 d y As we know, ∫ y n d y = y n + 1 n + 1 + c Area of A B C D = a 1 3 y 2 3 + 1 2 3 + 1 a 2 a = a 1 3 y 5 3 5 3 a 2 a = 3 5 a 1 3 y 5 3 a 2 a = 3 5 a 1 3 2 a 5 3 - a 5 3 = 3 5 a 1 3 a 5 3 2 5 3 - 1 = 3 5 a 1 3 + 5 3 2 5 3 - 1 = 3 5 a 2 2 5 3 - 1

E M B I B E

Understanding ISC Mathematics Class 12 Volume 2>Chapter 3 - Applications of Integrals>EXERCISE>Q 1

HARD

12th ICSE

IMPORTANT

Earn 100

Find the area of the region bounded by $y^{2} = 9 x, x = 2, x = 4$ and the $x - a x i s$ in the first quadrant.

Important Questions on Applications of Integrals

HARD

12th ICSE

IMPORTANT

Understanding ISC Mathematics Class 12 Volume 2>Chapter 3 - Applications of Integrals>EXERCISE>Q 2

Find the area of the region bounded by x² = 4y, y = 2, y = 4 and the y-axis in the first quadrant.

HARD

12th ICSE

IMPORTANT

Understanding ISC Mathematics Class 12 Volume 2>Chapter 3 - Applications of Integrals>EXERCISE>Q 2

Find the area of the region bounded by the curve ay² = x³, the y-axis and the lines y= a and y = 2a.

HARD

12th ICSE

IMPORTANT

Understanding ISC Mathematics Class 12 Volume 2>Chapter 3 - Applications of Integrals>EXERCISE>Q 3

Draw a rough sketch of the curve $y^{2} = 8 x$ and find the area of the region bounded by the curve $y^{2} = 8 x$ and the line $x = 2$ .

HARD

12th ICSE

IMPORTANT

Understanding ISC Mathematics Class 12 Volume 2>Chapter 3 - Applications of Integrals>EXERCISE>Q 4

If the area of the region bounded by the line $2 y = 5 x + 7$ , $x$ -axis and the lines $x = 2 and x = 8$ is $k$ sq. units, then the value of $k$ is

HARD

12th ICSE

IMPORTANT

Understanding ISC Mathematics Class 12 Volume 2>Chapter 3 - Applications of Integrals>EXERCISE>Q 5

Calculate the area under the curve $y = 2 \sqrt{x}$ included between the lines $x = 0$ and $x = 1$ . Write the answer without unit.

HARD

12th ICSE

IMPORTANT

Understanding ISC Mathematics Class 12 Volume 2>Chapter 3 - Applications of Integrals>EXERCISE>Q 6

Draw a rough sketch of the curve $y = \sqrt{x - 1}$ in the interval $[1, 5]$ . Find the area under the curve and between the lines $x = 1$ and $x = 5$ .