Grade 12 : Mathematics | Application of Integrals

Question 1

Find the area bounded by the parabola $y^2 = 8x$ and the line $x = 2$.

Accepted Answer

B) $\frac{32}{3} sq units

Answer

A) $\frac{16}{3} sq units

Answer

C) $\frac{64}{3} sq units

Answer

D) $16 sq units

Question 2

What is the area of the region bounded by the ellipse $\frac{x^2}{16} + \frac{y^2}{9} = 1$ in the first quadrant?

Accepted Answer

A) $3\pi sq units

Answer

B) $6\pi sq units

Answer

C) $12\pi sq units

Answer

D) $\frac{9\pi}{4} sq units

Question 3

Find the area bounded by the curve $y = \cos x$ between $x = 0$ and $x = \pi$ with the $x$-axis.

Accepted Answer

C) $2 sq units

Answer

A) $0 sq units

Answer

B) $1 sq unit

Answer

D) $4 sq units

Question 4

A civil engineer models the cross-section of a drainage canal using the curve $x^2 = 4y$ bounded above by the horizontal line $y = 4$.

Calculate the cross-sectional area of the canal.

Accepted Answer

A) $\frac{64}{3} sq units

Answer

B) $\frac{32}{3} sq units

Answer

C) $\frac{128}{3} sq units

Answer

D) $16 sq units

Question 5

Assertion (A): The area enclosed by the circle $x^2 + y^2 = a^2$ is $\pi a^2$.
Reason (R): The area is obtained by evaluating $4\int_0^a \sqrt{a^2 - x^2}\,dx$.

Accepted Answer

B) Both (A) and (R) are true, and (R) is the correct explanation of (A).

Answer

A) Both (A) and (R) are true, and (R) is not the correct explanation of (A).

Answer

C) (A) is true, but (R) is false.

Answer

D) (A) is false, but (R) is true.

Question 6

The area enclosed by the curve $y = x^3$, the $x$-axis, and the ordinates $x = -1$ and $x = 2$ is:

Accepted Answer

C) $\frac{17}{4} sq units

Answer

A) $\frac{15}{4} sq units

Answer

B) $4 sq units

Answer

D) $\frac{9}{2} sq units

Question 7

What is the area of the region bounded by the line $y = 3x + 2$, the $x$-axis, and the lines $x = -1$ and $x = 1$?

Accepted Answer

B) $\frac{25}{6} sq units

Answer

A) $4 sq units

Answer

C) $\frac{13}{3} sq units

Answer

D) $\frac{9}{2} sq units

Question 8

Find the area bounded by the curve $y = \sqrt{16 - x^2}$ and the $x$-axis.

Accepted Answer

B) $8\pi sq units

Answer

A) $16\pi sq units

Answer

C) $4\pi sq units

Answer

D) $2\pi sq units

Question 9

The area bounded by the curve $y = x|x|$, the $x$-axis, and the lines $x = -2$ and $x = 2$ is:

Accepted Answer

D) $\frac{16}{3} sq units

Answer

A) $0 sq units

Answer

B) $\frac{8}{3} sq units

Answer

C) $4 sq units

Question 10

A solar reflector's profile is designed along the parabola $y^2 = 12x$. A support bracket is placed vertically at its focal line.

What is the area bounded by the parabola and its latus rectum?

Accepted Answer

A) $24 sq units

Answer

B) $12 sq units

Answer

C) $48 sq units

Answer

D) $18 sq units

Question 11

Assertion (A): The area bounded by $y = e^x$, the $x$-axis, and the ordinates $x = 0$ and $x = 1$ is $e - 1$.
Reason (R): $\int_0^1 e^x\,dx = e^1 - e^0 = e - 1$.

Accepted Answer

C) Both (A) and (R) are true, and (R) is the correct explanation of (A).

Answer

A) (A) is true, but (R) is false.

Answer

B) Both (A) and (R) are true, but (R) is not the correct explanation of (A).

Answer

D) (A) is false, but (R) is true.

Question 12

Find the area bounded by the curve $y = 4 - x^2$ and the $x$-axis.

Accepted Answer

B) $\frac{32}{3} sq units

Answer

A) $\frac{16}{3} sq units

Answer

C) $\frac{8}{3} sq units

Answer

D) $16 sq units

Question 13

The line $x = c$ divides the area bounded by the parabola $y^2 = 4x$ and the line $x = 4$ into two equal parts. What is the value of $c$?

Accepted Answer

D) $4^{2/3}

Answer

A) $2

Answer

B) $2\sqrt{2}

Answer

C) $4^{1/3}

Question 14

Find the area bounded by the curve $y = \sin x$ between $x = 0$ and $x = 2\pi$ with the $x$-axis.

Accepted Answer

A) $4 sq units

Answer

B) $2 sq units

Answer

C) $0 sq units

Answer

D) $8 sq units

Question 15

A highway overpass is designed with an arched profile given by $y = 6x - x^2$ where the ground level lies along the line $y = 0$.

Calculate the vertical cross-sectional area under the arch.

Accepted Answer

B) $36 sq units

Answer

A) $18 sq units

Answer

C) $27 sq units

Answer

D) $72 sq units

Question 16

Assertion (A): The area of the region bounded by $y = |x|$ and $y = 2$ is 4 sq units.
Reason (R): The bounded region is a triangle with base 4 and height 2.

Accepted Answer

C) Both (A) and (R) are true, and (R) is the correct explanation of (A).

Answer

A) (A) is false, but (R) is true.

Answer

B) Both (A) and (R) are true, but (R) is not the correct explanation of (A).

Answer

D) (A) is true, but (R) is false.

Question 17

Find the area bounded by the curve $y = \frac{1}{x}$, the $x$-axis, and the ordinates $x = 1$ and $x = e^2$.

Accepted Answer

A) $2 sq units

Answer

B) $1 sq unit

Answer

C) $e^2 - 1 sq units

Answer

D) $\frac{1}{2} sq units

Question 18

The area enclosed by the parabola $x^2 = 2y$ and the line $y = 2$ is:

Accepted Answer

D) $\frac{16}{3} sq units

Answer

A) $\frac{8}{3} sq units

Answer

B) $4 sq units

Answer

C) $\frac{32}{3} sq units

Question 19

Find the area of the region bounded by $y = \sqrt{x}$, the $y$-axis, and the lines $y = 1$ and $y = 3$.

Accepted Answer

B) $\frac{26}{3} sq units

Answer

A) $\frac{26}{5} sq units

Answer

C) $\frac{13}{3} sq units

Answer

D) $8 sq units

Question 20

What is the area of the smaller region bounded by the circle $x^2 + y^2 = 4$ and the line $x = 1$?

Accepted Answer

C) $\frac{4\pi}{3} - \sqrt{3} sq units

Answer

A) $\frac{4\pi}{3} - \sqrt{3} sq units

Answer

B) $\frac{2\pi}{3} - \frac{\sqrt{3}}{2} sq units

Answer

D) $\frac{2\pi}{3} + \sqrt{3} sq units

Set 1Grade 12 : Mathematics | Application of Integrals