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Application of Integrals Set-6 MCQ Online Practice Test | Class 12 | Quiz Tweek

Application of Integrals Set-6

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

“Application of Integrals” is crucial because it links integral calculus to real geometry and physical quantities like area, volume, centroid and surface density. Board exams test direct computation of area/volume using correct limits and formulas, while competitive exams emphasize faster method selection (symmetry, substitution, washers vs shells) and careful setup.

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

(A)

$2$

(B)

$\dfrac{8}{3}$

(C)

$\dfrac{4}{3}$

(D)

$\dfrac{1}{3}$

Q2. Find the area of the region enclosed by the curves $x=y^2$ and $y=x^2$ .

(A)

$\dfrac{1}{3}$

(B)

$\dfrac{1}{2}$

(C)

$\dfrac{1}{4}$

(D)

$\dfrac{2}{3}$

Q3. The region $R$ is bounded by the curves $y=\sin x$ and $y=\cos x$ for $0\le x\le \dfrac{\pi}{2}$ . The area of $R$ is

(A)

$\sqrt{2}-1$

(B)

$1$

(C)

$2\sqrt{2}$

(D)

$2\bigl(\sqrt{2}-1\bigr)$

Q4. Assertion (A): If $f$ is continuous on $[a,b]$ and $f(x)+f(a+b-x)=C$ for all $x\in[a,b]$ (where $C$ is a constant), then $\displaystyle\int_a^b f(x)\,dx=\dfrac{C(b-a)}{2}$ .
Reason (R): Making the substitution $u=a+b-x$ in the integral gives $\displaystyle\int_a^b f(a+b-x)\,dx=\int_a^b f(x)\,dx$ , and adding the two integrals yields $2\int_a^b f(x)\,dx=\int_a^b C\,dx$ .

(A)

Both A and R are true but R does not correctly explain A.

(B)

Both A and R are true and R correctly explains 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 $0\le x\le1$ is rotated about the vertical line $x=1$ . The volume of the solid obtained is

(A)

$\dfrac{\pi}{6}$

(B)

$\dfrac{\pi}{3}$

(C)

$\dfrac{\pi}{12}$

(D)

$\dfrac{\pi}{4}$

Q6. The curve $y=x^3$ and its tangent at $x=1$ enclose a region. Find the area of this region.

(A)

$\dfrac{9}{2}$

(B)

$\dfrac{27}{8}$

(C)

$\dfrac{27}{4}$

(D)

$\dfrac{9}{4}$

Q7. The region bounded by the curves $y=x^2$ and $y=2x$ (for their points of intersection) is revolved about the vertical line $x=1$ . The volume of the solid generated is

(A)

$\pi$

(B)

$2\pi$

(C)

$\dfrac{\pi}{2}$

(D)

$\dfrac{3\pi}{2}$

Q8. Let $R$ be the region bounded by $y=\sqrt{x}$ and $y=x^2$ for $0\le x\le1$ . The $x$ -coordinate of the centroid of $R$ is

(A)

$\dfrac{3}{10}$

(B)

$\dfrac{7}{18}$

(C)

$\dfrac{1}{3}$

(D)

$\dfrac{9}{20}$

Q9. Find the area of the region enclosed by the parabola $y^2=4x$ and the circle $x^2+y^2=8x$ .

(A)

$8\pi-\dfrac{32}{3}$

(B)

$\dfrac{24\pi-64}{3}$

(C)

$12\pi-\dfrac{64}{3}$

(D)

$\dfrac{8\pi-64}{3}$

Q10. Compute the area of the region between the curves $y=\dfrac{1}{x}$ and $y=\dfrac{1}{x+1}$ for $x\ge 1$ (i.e., the area between them from $x=1$ to $x\to\infty$ ).

(A)

$\ln 2$

(B)

$\dfrac{1}{2}\ln 2$

(C)

$\ln\!\dfrac{3}{2}$

(D)

$1$

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

(A)

$\dfrac{1}{3}$

(B)

$4$

(C)

$\dfrac{4}{3}$

(D)

$\dfrac{8}{3}$

Q12. The region bounded by the curves $y=x^2$ and $y=2x-x^2$ is taken as a lamina of uniform density. Find the $x$ -coordinate of its centroid.

(A)

$\dfrac{1}{3}$

(B)

$\dfrac{1}{2}$

(C)

$\dfrac{2}{3}$

(D)

$\dfrac{3}{4}$

Q13. Find the total area enclosed between the curves $y=x^3$ and $y=x$ .

(A)

$\dfrac{1}{4}$

(B)

$\dfrac{3}{4}$

(C)

$1$

(D)

$\dfrac{1}{2}$

Q14. Assertion (A): The area enclosed between the curves $y=e^x$ and $y=x+1$ is zero.
Reason (R): The function $f(x)=e^x-x-1$ has a unique minimum at $x=0$ and $f(0)=0$ .

(A)

Both (A) and (R) are true and (R) explains (A).

(B)

Both (A) and (R) are true but (R) does not explain (A).

(C)

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

(D)

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

Q15. A thin lamina occupies the region bounded by $y=x^2$ and $y=x$ for $0\le x\le1$ . Its surface density at any point is proportional to the distance from the $x$ -axis. Find the $x$ -coordinate of the centroid of the lamina.

(A)

$\dfrac{1}{2}$

(B)

$\dfrac{5}{8}$

(C)

$\dfrac{3}{4}$

(D)

$\dfrac{2}{3}$

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