Find the area of the region bounded by the curve $4x^2 + y^2 = 36$ using integration.
CBSE CLASS 12 • MATHEMATICS • CHAPTER 08
Applications of the Integrals
51 distinct questions across 72 board paper set appearances (2022–2026).
The area of the region bounded by the curve y = x and x-axis, between x = 0 and x = 2 is : (A) 2 sq. units (B) $\frac{1}{2}$ sq. unit (C) 1 sq. unit (D) 4 sq. units
(a) Using integration, find the area bounded by the curve $y = x^2$ and the line $y = 2$.
OR
(b) Using integration, find the area bounded by the curve $9x^2 + 16y^2 = 144$ and the lines $x = 2$, $x = -2$.
Calculate the area of the region bounded by the curve $\frac{x^2}{9} + \frac{y^2}{4} = 1$ and the x-axis using integration.
The area bounded by the curve $y = x|x|$, $x$-axis and the ordinates $x = -1$ and $x = 1$ is given by
(A) $0$
(B) $\frac{1}{3}$
(C) $\frac{2}{3}$
(D) $\frac{4}{3}$
The area of the region bounded by the curve $y^2 = x$ between $x = 0$ and $x = 1$ is :
(A) $\frac{3}{2} \text{ sq units}$
(B) $\frac{2}{3} \text{ sq units}$
(C) $3 \text{ sq units}$
(D) $\frac{4}{3} \text{ sq units}$
(a) Sketch the graph of $y = x|x|$ and hence find the area bounded by this curve, X-axis and the ordinates $x = -2$ and $x = 2$, using integration. OR (b) Using integration, find the area bounded by the ellipse $9x^2 + 25y^2 = 225$, the lines $x = -2$, $x = 2$, and the X-axis.
Area of the region bounded by curve $y^2 = 4x$ and the X-axis between $x = 0$ and $x = 1$ is : (A) $\frac{2}{3}$ (B) $\frac{8}{3}$ (C) $3$ (D) $\frac{4}{3}$
The area of the region bounded by the curve $y^{2}=4x$ and $x=1$ is :
(A) $\frac{4}{3}$
(B) $\frac{8}{3}$
(C) $\frac{64}{3}$
(D) $\frac{32}{3}$
The area bounded by the curve $y = \sqrt{x}$, Y-axis and between the lines $y = 0$ and $y = 3$ is :
(A) $2\sqrt{3}$
(B) 27
(C) 9
(D) 3
(a) Using integration, find the area of the region bounded by the curve y = \sqrt{4 - x^2}, the lines x = -\sqrt{2} and x = \sqrt{3} and the x-axis. OR (b) Using integration, evaluate the area of the region bounded by the curve y = x^2, the lines y = 1 and y = 3 and the y-axis.
Using integration, find the area of the region bounded by the parabola $y^2 = 4ax$ and its latus rectum.
(b) If the area of the region bounded by the curve $y^2 = 4ax$ and the line $x = 4a$ is $\frac{256}{3}$ sq. units, then using integration, find the value of $a$, where $a > 0$.
(b) Using integration, find the area of the region bounded by the parabola y^2 = 4x, the lines x = 0 and x = 3 and the x-axis.
The area bounded by the curve $y = x|x|$, $x$-axis and the ordinates $x = -1$ and $x = 1$ is given by
(A) $0$
(B) $\\frac{1}{3}$
(C) $\\frac{2}{3}$
(D) $\\frac{4}{3}$
Find the area of the region bounded by the line $y = 3x + 2$, the x-axis and the ordinates $x = -1$ and $x = 1$.
Using integration, find the area of the region bounded by the curve $y = x|x|$, x-axis, $x = - 2$ and $x = 2$.
Using integration, find the area of the region bounded by $y = 5x + 4$, $y = 0$, $x = -1$ and $x = 1$.
Using integration, find the area of the region bounded by the ellipse $\frac{x^2}{4} + \frac{y^2}{3} = 1$ and the lines $x = -\sqrt{2}$ and $x = \sqrt{2}$.
The area enclosed by the curve $y = \sqrt{4 - x^2}$ and the coordinate axes in the first quadrant is :
(A) $2\pi \text{ sq. units}$
(B) $\pi \text{ sq. units}$
(C) $\frac{\pi}{2} \text{ sq. units}$
(D) $4\pi \text{ sq. units}$
Using integration, find the area of the region bounded between the lines $x = -2$, $x = 2$ and the circle $x^2 + y^2 = 16$.
The area bounded by the parabola $x^2 = y$ and the line $y = 1$ is :
(A) $\frac{2}{3} \text{ sq unit}$
(B) $\frac{1}{3} \text{ sq unit}$
(C) $\frac{4}{3} \text{ sq units}$
(D) $2 \text{ sq units}$
The area bounded by the curve $x = y^2$, y-axis and the lines y = 3 and y = 4 is :
(A) $\frac{74}{3}$ sq units
(B) $\frac{37}{3}$ sq units
(C) 74 sq units
(D) 37 sq units
The area bounded by the line $y = x$, $y$-axis and the lines $y = 0$ and $y = 4$ is :
(A) $16 \text{ sq units}$
(B) $8 \text{ sq units}$
(C) $4 \text{ sq units}$
(D) $2 \text{ sq units}$
Using integration, find the area of the region bounded by the line $y = 5x + 2$, the $x$-axis and the ordinates $x = -2$ and $x = 2$.
Using integration, find the area of the region bounded by the line $y = 5x + 2$, the $x-\text{axis}$ and the ordinates $x = -2$ and $x = 2$.
The area of the region bounded by the lines $y = x+1$, $x=1$, $x=3$ and $x$-axis is
(A) $6 \text{ sq units}$
(B) $8 \text{ sq units}$
(C) $7.5 \text{ sq units}$
(D) $2 \text{ sq units}$
Sketch a graph of $y = x^2$. Using integration, find the area of the region bounded by $y = 9$, $x = 0$ and $y = x^2$.
Draw a rough sketch of the curve $y = \sqrt{x}$. Using integration, find the area of the region bounded by the curve $y = \sqrt{x}$, $x = 4$ and x-axis, in the first quadrant.
Draw a rough sketch for the curve $y = 2 + |x + 1|$. Using integration, find the area of the region bounded by the curve $y = 2 + |x + 1|$, $x = -4$, $x = 3$ and $y = 0$.
The area of the region bounded by the curve $y^2 = x$ between $x = 0$ and $x = 1$ is :
(A) $\frac{3}{2}$ sq units
(B) $\frac{2}{3}$ sq units
(C) $3$ sq units
(D) $\frac{4}{3}$ sq units
Draw the rough sketch of the curve $y = 20 \cos 2x$; (where $\frac{\pi}{6} \leq x \leq \frac{\pi}{3}$). Using integration, find the area of the region bounded by the curve $y = 20 \cos 2x$ from the ordinates $x = \frac{\pi}{6}$ to $x = \frac{\pi}{3}$ and the $x$-axis.
Using integration, find the area of the region bounded by the curve $y = x^2, x = - 1, x = 1$ and the x-axis.
Find the area of the region bounded by the lines $x - 2y = 4$, $x = -1$, $x = 6$ and x-axis, using integration.
Find the area of the region bounded by the curves $x^2 = y$, $y = x + 2$ and x-axis, using integration.
Find the area of the region bounded by the lines $y = 4x + 5$, $x + y = 5$ and $4y = x + 5$, using integration.
Find the area of the region bounded by the curve y = \sqrt{x}, the line x = 2y + 3 and the x-axis, using integration.
The area of the region bounded by the line y = mx (m > 0), the curve x^{2} + y^{2} = 4 and the x-axis in the first quadrant is \frac{π}{2} units. Using integration, find the value of m.
The area of the region bounded by the line $y = mx$ $(m > 0)$, the curve $x^{2} + y^{2} = 4$ and the x-axis in the first quadrant is $\frac{\pi}{2}$ units. Using integration, find the value of m.
Using integration, find the area of the region bounded by the circle $x^2 + y^2 = 16$, line $y = x$ and $y$-axis, but lying in the $1^{ ext{st}}$ quadrant.
Using integration, find the area of the region bounded by the triangle ABC when its sides are given by the lines $4x - y + 5 = 0$, $x + y - 5 = 0$ and $x - 4y + 5 = 0$.
Using integration, find the area of the region bounded by the triangle whose vertices are $(-1, 2)$, $(1, 5)$ and $(3, 4)$.
Using integration, find the area enclosed by the curve $\frac{x^2}{9} + \frac{y^2}{4} = 1$.
(a) Using integration, find the area of the region $\{(x, y) : 4x^2 + 9y^2 \leq 36, 2x + 3y \geq 6\}$.
OR
(b) Using integration, find the area of the region bounded by lines $x - y + 1 = 0$, $x = -2$, $x = 3$ and $x$-axis.
(a) Using integration, find the area of the region $\{(x, y) : 4x^2 + 9y^2 \le 36, 2x + 3y \ge 6\}$.
OR
(b) Using integration, find the area of the region bounded by lines $x - y + 1 = 0$, $x = -2$, $x = 3$ and $x$-axis.
(a) Using integration, find the area of the smaller region enclosed by the curve $4x^2 + 4y^2 = 9$ and the line $2x + 2y = 3$. OR (b) If the area of the region bounded by the curve $y^2 = 4ax$ and the line $x = 4a$ is $\frac{256}{3}$ sq. units, then using integration, find the value of a, where $a >...
Find the area of the region bounded by curve $4x^2 = y$ and the line $y = 8x + 12$, using integration.
Using integration, find the area of the region bounded by the curves $x^2 + y^2 = 4$, $x = \sqrt{3} y$ and x-axis lying in the first quadrant.
If the area of the region bounded by the line $y = mx$ and the curve $x^2 = y$ is $\frac{32}{3}$ sq. units, then find the positive value of $m$, using integration.
Find the area bounded by the curves $y = |x-1|$ and $y = 1$, using integration.
(a) Find the area of the region $\{(x, y) : x^2 + y^2 \leq 9, x + y \geq 3\}$, using integration.
OR
(b) Using integration, find the area of the region bounded by the parabola $y^2 = 4x$, the lines $x = 0$ and $x = 3$ and the x-axis.