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Application of Integrals Set-5 MCQ Online Practice Test | Class 12 | Quiz Tweek

Application of Integrals Set-5

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

The chapter “Application of Integrals” is crucial for Class 12 board and competitive exams because it links integral calculus with real geometric quantities—areas, volumes of solids of revolution, centroid and moments, and related convergence/divergence results. Mastering standard integral setups (limits, washers/shells) and interpreting results (finite/infinite) helps score reliably in both theory and numerical problems.

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Q1. Find the area of the region enclosed by the curves $y=x^2$ and $y=2x+3$ .

(A)

$\dfrac{28}{3}$

(B)

$16$

(C)

$\dfrac{32}{3}$

(D)

$\dfrac{34}{3}$

Q2. The region $R$ is bounded by the parabola $x=y^2$ and the line $x=4y+4$ . The area of $R$ is

(A)

$\dfrac{64}{3}$

(B)

$\dfrac{64\sqrt{2}}{3}$

(C)

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

(D)

$\dfrac{128\sqrt{2}}{9}$

Q3. The region bounded by $y=x$ and $y=x^2$ for $0\le x\le 1$ is revolved about the line $y=1$ . The volume of the solid of revolution is

(A)

$\dfrac{\pi}{5}$

(B)

$\dfrac{\pi}{15}$

(C)

$\dfrac{2\pi}{15}$

(D)

$\dfrac{4\pi}{15}$

Q4. Let $f$ be continuous and invertible on $[a,b]$ with $f(a)=c$ and $f(b)=d$ . Consider the statements:
Statement A: $\displaystyle \int_a^b f(x)\,dx + \int_c^d f^{-1}(y)\,dy = b\,d - a\,c.$
Statement R: The identity follows by the substitution $y=f(x)$ in the second integral giving $\displaystyle \int_c^d f^{-1}(y)\,dy = \big[ x\,f(x)\big]_a^b - \int_a^b f(x)\,dx$ .

(A)

Both A and R are true but R is not a correct explanation of A.

(B)

Both A and R are false.

(C)

A is true but R is false.

(D)

Both A and R are true and R is a correct explanation of A.

Q5. Find the area of the region enclosed by the parabola $y^2=4x$ and the line $x+y=2$ .

(A)

$4\sqrt{3}$

(B)

$8\sqrt{3}$

(C)

$12$

(D)

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

Q6. Find the area of the region bounded by the curve $y = x^{2} - 4x + 3$ and the $x$ -axis.

(A)

$2$

(B)

$\dfrac{1}{3}$

(C)

$\dfrac{4}{3}$

(D)

$\dfrac{8}{3}$

Q7. The region enclosed by the curves $y = x^{2}$ and $x = y^{2}$ in the first quadrant has area equal to

(A)

$\dfrac{1}{3}$

(B)

$\dfrac{1}{2}$

(C)

$\dfrac{1}{6}$

(D)

$\dfrac{2}{3}$

Q8. Let $R$ be the region in the first quadrant bounded by $y=x^{2}$ and $y=\sqrt{x}$ for $0\le x\le1$ . If $V_x$ is the volume obtained by revolving $R$ about the $x$ -axis and $V_y$ the volume obtained by revolving $R$ about the $y$ -axis, then

(A)

$V_x > V_y$

(B)

$V_x < V_y$

(C)

$V_x = V_y = \dfrac{\pi}{5}$

(D)

$V_x = V_y = \dfrac{3\pi}{10}$

Q9. For the region under the curve $y=e^{-x}$ for $x\in[0,\infty)$ (above the $x$ -axis), the $x$ -coordinate of the centroid $x_c$ is

(A)

$\dfrac{1}{2}$

(B)

$1$

(C)

$2$

(D)

$\infty$

Q10. Assertion (A): The area under the curve $y=\dfrac{1}{x}$ over $[1,\infty)$ is infinite. Reason (R): The volume generated by revolving the same region about the $x$ -axis is finite.

(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.

Q11. Find the area of the region enclosed by the curves $y=\sqrt{x}$ and $y=\dfrac{x}{2}$ .

(A)

$\dfrac{8}{3}$

(B)

$2$

(C)

$\dfrac{4}{3}$

(D)

$\dfrac{5}{3}$

Q12. The region bounded by the curves $y=x^2$ and $y=2x$ is revolved about the $x$ -axis. The volume of the solid generated is

(A)

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

(B)

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

(C)

$\dfrac{64\pi}{5}$

(D)

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

Q13. The area of the region enclosed by the parabola $y^2=4x$ and the line $x=y+2$ is

(A)

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

(B)

$4\sqrt{3}$

(C)

$12\sqrt{3}$

(D)

$8\sqrt{3}$

Q14. Assertion (A): For a smooth positive function $y=f(x)$ on $[a,b]$ , the surface area of the solid generated by revolving the curve about the $x$ -axis is given by

S=2\pi\int_a^b f(x)\sqrt{1+\bigl(f'(x)\bigr)^2}\,dx.

Reason (R): For any constant $k>0$ , the surface area of the solid generated by revolving $y=kf(x)$ about the $x$ -axis equals $k$ times the surface area generated by $y=f(x)$ .

(A)

Both A and R are true and R is a correct explanation of A

(B)

A is true, R is false

(C)

Both A and R are true but R is not a correct explanation for A

(D)

A is false, R is true

Q15. For which values of $p>0$ does the solid obtained by revolving the region bounded by $y=\dfrac{1}{x^p}$ , the $x$ -axis and the vertical line $x=1$ (for $x\to\infty$ ) about the $x$ -axis have finite volume but infinite surface area?

(A)

$\left(\tfrac{1}{2},\,1\right]$

(B)

$p>1$

(C)

$0<p\le\tfrac{1}{2}$

(D)

$p=1$ only

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