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Application of Integrals Set-2 MCQ Online Practice Test | Class 12 | Quiz Tweek

Application of Integrals Set-2

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

The chapter “Application of Integrals” is crucial because it builds the core techniques used in CBSE board and competitive exams to compute areas, volumes, centroids, and other physical quantities. It connects integration with geometry—so mastering these standard setups and careful bounds directly improves accuracy and speed in problems across JEE and NEET.

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Q1. Find the area enclosed by the curves $y=x^2$ and $y=2x$ .

(A)

$\dfrac{8}{3}$

(B)

$\dfrac{2}{3}$

(C)

$\dfrac{4}{3}$

(D)

$\dfrac{16}{3}$

Q2. The region bounded by $y=x^2$ and $y=2x$ for $0\le x\le 2$ is revolved about the $x$ -axis. The volume of the solid generated is:

(A)

$\dfrac{64\pi}{15}$

(B)

$\dfrac{32\pi}{15}$

(C)

$\dfrac{64\pi}{45}$

(D)

$\dfrac{128\pi}{15}$

Q3. Find the area of the region enclosed by the curves $x=y^2$ and $x=4-y^2$ .

(A)

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

(B)

$\dfrac{32\sqrt{2}}{3}$

(C)

$\dfrac{16}{3}$

(D)

$\dfrac{16\sqrt{2}}{3}$

Q4. The area enclosed between the curves $y=\sin x$ and $y=\cos x$ on the interval $[0,\pi]$ equals:

(A)

$2$

(B)

$2\sqrt{2}$

(C)

$\sqrt{2}$

(D)

$4\sqrt{2}$

Q5. Let $R$ be the region bounded by $y=x^2$ and $y=4x-x^2$ . The centroid (center of mass) of $R$ is:

(A)

$(1,2)$

(B)

$\left(1,\dfrac{3}{2}\right)$

(C)

$\left(\dfrac{3}{2},2\right)$

(D)

$\left(\dfrac{4}{3},\dfrac{11}{6}\right)$

Q6. Find the area of the region enclosed by the curves $y=x^2$ and $y=2x-x^2$ .

(A)

$\frac{1}{2}$

(B)

$\frac{1}{6}$

(C)

$\frac{1}{3}$

(D)

$\frac{2}{3}$

Q7. The parabola $y^2=4x$ and the line $y=2x-2$ meet in two points. Find the area of the finite region bounded by these curves.

(A)

$\displaystyle \frac{5\sqrt{5}}{3}$

(B)

$\displaystyle \frac{10}{3}$

(C)

$\displaystyle \frac{5\sqrt{5}}{6}$

(D)

$\displaystyle \frac{5}{3}$

Q8. Compute the total area enclosed between the curve $y=x^3-x$ and the $x$ -axis on the interval $[-1,1]$ .

(A)

$\frac{1}{4}$

(B)

$\frac{1}{3}$

(C)

$1$

(D)

$\frac{1}{2}$

Q9. The region bounded by $y=x^2$ and $y=\sqrt{x}$ for $0\le x\le 1$ is rotated about the horizontal line $y=2$ . Find the volume of the solid generated.

(A)

$\displaystyle \frac{31\pi}{60}$

(B)

$\displaystyle \frac{31\pi}{30}$

(C)

$\displaystyle \frac{7\pi}{6}$

(D)

$\displaystyle \frac{\pi}{3}$

Q10. Let $a>0$ be the positive intersection (other than $x=0$ ) of the curves $y=\sqrt{x}$ and $y=kx^2$ . If the area enclosed between them from $x=0$ to $x=a$ equals half the area enclosed between $y=\sqrt{x}$ and $y=x^2$ from $x=0$ to $x=1$ , find the value of $k$ .

(A)

$2$

(B)

$\dfrac{1}{2}$

(C)

$\dfrac{3}{2}$

(D)

$1$

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