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Class 12 Mathematics — Application of Integrals (MCQ Set-1) MCQ Online Practice Test | Class 12 | Quiz Tweek

Class 12 Mathematics — Application of Integrals (MCQ Set-1)

Practice Important MCQs for Application of Integrals — Mathematics, Class 12.

The chapter “Application of Integrals” is crucial in both board exams and competitive tests because it directly connects integration with real geometric applications—areas between curves, volumes of solids of revolution, and centroids (center of mass) for laminae. Mastery of setting up correct integrals (choice of limits, correct “washer/disk” geometry, and centroid formulas) ensures high accuracy and speed in problem-solving.

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Q1. Find the area of the region enclosed by the curves $y=\sqrt{x}$ and $y=x$ .

(A)

$\dfrac{1}{3}$

(B)

$\dfrac{1}{4}$

(C)

$\dfrac{1}{6}$

(D)

$\dfrac{1}{12}$

Q2. The region in the first quadrant bounded by $y=x$ and $y=x^2$ is revolved about the $y$ -axis. The volume of the solid generated is

(A)

$\dfrac{\pi}{6}$

(B)

$\dfrac{\pi}{3}$

(C)

$\dfrac{\pi}{12}$

(D)

$\dfrac{\pi}{8}$

Q3. Find the total area enclosed between the curve $y=x^3-3x$ and the $x$ -axis (between its three real $x$ -intercepts).

(A)

$\dfrac{9}{4}$

(B)

$\dfrac{9}{2}$

(C)

$3$

(D)

$\dfrac{27}{4}$

Q4. Assertion (A): For a continuous function $f$ with $f(x)\ge 0$ on $[a,b]$ , the volume obtained by rotating the region between $y=f(x)$ and the $x$ -axis about the $x$ -axis equals $\pi\displaystyle\int_a^b f(x)\,dx$ .
Reason (R): By the disk method, each cross-section perpendicular to the $x$ -axis is a circle of radius $f(x)$ , so its area is $\pi[f(x)]^2$ and the volume is $\pi\displaystyle\int_a^b[f(x)]^2\,dx$ .

(A)

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

(B)

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

(C)

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

(D)

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

Q5. The region bounded by the curves $y=x$ and $y=x^2$ for $0\le x\le 1$ is revolved about the line $y=-1$ . The volume of the solid generated is

(A)

$\dfrac{7\pi}{15}$

(B)

$\dfrac{7\pi}{30}$

(C)

$\dfrac{\pi}{3}$

(D)

$\dfrac{2\pi}{5}$

Q6. The area of the region enclosed by the curves $y=x^2+x$ and $y=x+2$ is

(A)

( $\dfrac{4\sqrt{2}}{3}$ )

(B)

( $\dfrac{8}{3}$ )

(C)

( $\dfrac{8\sqrt{2}}{3}$ )

(D)

( $\dfrac{16\sqrt{2}}{9}$ )

Q7. The region bounded by the curves $y=\sqrt{x}$ and $y=\dfrac{x}{2}$ between their points of intersection is revolved about the $x$ -axis. The volume of the solid generated is

(A)

( $\dfrac{8\pi}{3}$ )

(B)

( $\dfrac{16\pi}{3}$ )

(C)

( $\dfrac{4\pi}{3}$ )

(D)

( $\dfrac{8\pi}{9}$ )

Q8. The area of the region enclosed by the curves $y=x$ and $y=x^3$ is

(A)

( $\dfrac{1}{4}$ )

(B)

( $1$ )

(C)

( $\dfrac{1}{3}$ )

(D)

( $\dfrac{1}{2}$ )

Q9. A lamina occupies the region bounded by the curves $y=x^2$ and $y=2x$ (between their intersections at $x=0$ and $x=2$ ). If the surface density at a point $(x,y)$ is $\rho(x,y)=kx$ (with $k>0$ ), the $x$ -coordinate of the centre of mass equals

(A)

( $\dfrac{4}{3}$ )

(B)

( $\dfrac{6}{5}$ )

(C)

( $\dfrac{3}{2}$ )

(D)

( $\dfrac{5}{4}$ )

Q10. Consider the solid obtained by rotating the region under the curve $y=\dfrac{1}{x}$ for $x\in[1,\infty)$ about the $x$ -axis. Which of the following is true?

(A)

(It has finite volume equal to $\pi$ and infinite lateral surface area.)

(B)

(It has finite volume equal to $\pi$ and finite lateral surface area.)

(C)

(It has infinite volume and infinite lateral surface area.)

(D)

(It has infinite volume but finite lateral surface area.)

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