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Application of Integrals Set-3

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

The chapter “Application of Integrals” is crucial because it builds the core integral techniques used across Class 12 board exams and competitive tests—especially for finding areas, volumes of solids of revolution, and centroids. Mastery here strengthens your ability to set correct bounds, choose the right method (washers/shells), and perform integration accurately—skills directly tested in JEE/NEET-style math problems.

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Q1. Let $R$ be the region enclosed by the curves $y=x^2$ and $y=2x$ . What is the area of $R$ ?

(A)

$\dfrac{2}{3}$

(B)

$\dfrac{4}{3}$

(C)

$\dfrac{8}{3}$

(D)

$\dfrac{10}{3}$

Q2. Find the positive value of $a$ such that the area enclosed between the curves $y=x^2$ and $y=ax$ is $2$ square units.

(A)

$\sqrt[3]{6}$

(B)

$\sqrt[3]{3}$

(C)

$\sqrt[3]{24}$

(D)

$\sqrt[3]{12}$

Q3. The region bounded by $y=\sqrt{x}$ , $y=0$ and $x=4$ is revolved about the $y$ -axis. What is the volume of the solid formed?

(A)

$\dfrac{128\pi}{5}$

(B)

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

(C)

$\dfrac{256\pi}{5}$

(D)

$\dfrac{128\pi}{3}$

Q4. Assertion (A): The area enclosed by the curve $y=x^3-x$ and the $x$ -axis between $x=-1$ and $x=1$ is $0$ .
Reason (R): $\displaystyle \int_{-1}^{1}(x^3-x)\,dx=0$ because the integrand is an odd function.

(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 false but R is true.

(D)

Both A and R are false.

Q5. Let $R$ be the region bounded by $y=\dfrac{1}{x}$ , the $x$ -axis and the vertical line $x=1$ , for $x\ge 1$ . When $R$ is revolved, which statement is correct?

(A)

Revolution about both axes gives finite volumes, each equal to $\pi$ .

(B)

Revolution about the $x$ -axis gives finite volume $\pi$ , while revolution about the $y$ -axis gives infinite volume.

(C)

Revolution about the $x$ -axis gives infinite volume, while revolution about the $y$ -axis gives finite volume $\pi$ .

(D)

Revolutions about both axes give infinite volumes.

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

(A)

$\dfrac{1}{6}$

(B)

$\dfrac{1}{2}$

(C)

$\dfrac{1}{3}$

(D)

$\dfrac{2}{3}$

Q7. For the region bounded by $y=x^2$ and $y=4x-x^2$ , find the x-coordinate of the centroid of the region.

(A)

$1$

(B)

$\dfrac{4}{3}$

(C)

$\dfrac{2}{3}$

(D)

$\dfrac{3}{2}$

Q8. Compute the area of the region bounded by the parabola $y=x^2$ and the line $x+y=6$ .

(A)

$\dfrac{125}{12}$

(B)

$\dfrac{25}{6}$

(C)

$\dfrac{125}{3}$

(D)

$\dfrac{125}{6}$

Q9. A lamina occupies the region bounded by $y=x^2$ and $y=4x-x^2$ (between their intersection points). The density at any point is proportional to its distance from the y‑axis, i.e. $\rho(x,y)=k|x|$ . Find the x‑coordinate of the centroid of this lamina.

(A)

$1$

(B)

$\dfrac{6}{5}$

(C)

$\dfrac{3}{2}$

(D)

$\dfrac{7}{5}$

Q10. The curve $y=x^3-3x$ meets the x‑axis at $x=-\sqrt{3},0,\sqrt{3}$ . Find the area of the region bounded by the curve and the x‑axis between $x=-2$ and $x=2$ .

(A)

$5$

(B)

$\dfrac{7}{2}$

(C)

$4$

(D)

$10$

Q11. Find the area of the region enclosed by the curves $y=x$ and $y=x^3$ .

(A)

$\dfrac{1}{4}$

(B)

$\dfrac{1}{2}$

(C)

$1$

(D)

$\dfrac{3}{4}$

Q12. The curves $x=y^2$ and $y=x^2$ enclose a region in the first quadrant. Its area is

(A)

$\dfrac{1}{6}$

(B)

$\dfrac{1}{4}$

(C)

$\dfrac{1}{2}$

(D)

$\dfrac{1}{3}$

Q13. The region bounded by $y=\ln x$ , the $x$ -axis and the vertical lines $x=1$ and $x=e$ is revolved about the $x$ -axis. The volume of the solid of revolution is

(A)

$\pi\bigl(e-2\bigr)$

(B)

$\pi\bigl(e-1\bigr)$

(C)

$\pi\bigl(2-e\bigr)$

(D)

$\pi\bigl(e-3\bigr)$

Q14. Let $R$ be the region bounded by $y=x^2$ and $y=x$ for $0\le x\le1$ . The solid obtained by revolving $R$ about the line $y=-1$ has volume equal to

(A)

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

(B)

$\dfrac{\pi}{3}$

(C)

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

(D)

$\dfrac{\pi}{6}$

Q15. Consider the curve $y=\dfrac{x^2}{1+x^2}$ and its horizontal asymptote $y=1$ . The area of the region between the curve and the line $y=1$ from $x=0$ to $x\to\infty$ is

(A)

$1$

(B)

$\dfrac{\pi}{2}$

(C)

$\dfrac{\pi}{4}$

(D)

$\ln 2$

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