JEE Main
Mathematics
Area Under The Curves
Previous Years Questions

## Numerical

If the area bounded by the curve $2 y^{2}=3 x$, lines $x+y=3, y=0$ and outside the circle $(x-3)^{2}+y^{2}=2$ is $\mathrm{A}$, then $4(\pi+4 A)$ is eq...
If A is the area in the first quadrant enclosed by the curve $$\mathrm{C: 2 x^{2}-y+1=0}$$, the tangent to $$\mathrm{C}$$ at the point $$(1,3)$$ and t...
If the area of the region $$\left\{(x, \mathrm{y}):\left|x^{2}-2\right| \leq y \leq x\right\}$$ is $$\mathrm{A}$$, then $$6 \mathrm{A}+16 \sqrt{2}$$ i...
Let $$y = p(x)$$ be the parabola passing through the points $$( - 1,0),(0,1)$$ and $$(1,0)$$. If the area of the region $$\{ (x,y):{(x + 1)^2} + {(y -... Let the area enclosed by the lines$$x+y=2, \mathrm{y}=0, x=0$$and the curve$$f(x)=\min \left\{x^{2}+\frac{3}{4}, 1+[x]\right\}$$where$$[x]$$den... If the area of the region$$S=\left\{(x, y): 2 y-y^{2} \leq x^{2} \leq 2 y, x \geq y\right\}$$is equal to$$\frac{n+2}{n+1}-\frac{\pi}{n-1}$$, then t... Let$$A$$be the area bounded by the curve$$y=x|x-3|$$, the$$x$$-axis and the ordinates$$x=-1$$and$$x=2$$. Then$$12 A$$is equal to ____________... Let the area of the region \left\{(x, y):|2 x-1| \leq y \leq\left|x^{2}-x\right|, 0 \leq x \leq 1\right\} be \mathrm{A}. Then (6 \mathrm{~A}+11)^... Let for$$x \in \mathbb{R}$$,$$ f(x)=\frac{x+|x|}{2} \text { and } g(x)=\left\{\begin{array}{cc} x, & x Then area bounded by the curve $$y=(f \circ g... Let A be the area of the region \left\{(x, y): y \geq x^2, y \geq(1-x)^2, y \leq 2 x(1-x)\right\}. Then 540 \mathrm{~A} is equal to :... Let$$\alpha$$be the area of the larger region bounded by the curve$$y^{2}=8 x$$and the lines$$y=x$$and$$x=2$$, which lies in the first quadrant... If the area enclosed by the parabolas$$\mathrm{P_1:2y=5x^2}$$and$$\mathrm{P_2:x^2-y+6=0}$$is equal to the area enclosed by$$\mathrm{P_1}$$and$$...
If the area of the region bounded by the curves $$y^2-2y=-x,x+y=0$$ is A, then 8 A is equal to __________
Let the area enclosed by the x-axis, and the tangent and normal drawn to the curve $$4{x^3} - 3x{y^2} + 6{x^2} - 5xy - 8{y^2} + 9x + 14 = 0$$ at the p...
If for some $$\alpha$$ > 0, the area of the region $$\{ (x,y):|x + \alpha | \le y \le 2 - |x|\}$$ is equal to $${3 \over 2}$$, then the area of the r...
For real numbers a, b (a > b > 0), let Area $$\left\{ {(x,y):{x^2} + {y^2} \le {a^2}\,and\,{{{x^2}} \over {{a^2}}} + {{{y^2}} \over {{b^2}}} \ge 1} \r... If the area of the region$$\left\{ {(x,y):{x^{{2 \over 3}}} + {y^{{2 \over 3}}} \le 1,\,x + y \ge 0,\,y \ge 0} \right\}$$is A, then$${{256A} \over ...
Let $${A_1} = \left\{ {(x,y):|x| \le {y^2},|x| + 2y \le 8} \right\}$$ and $${A_2} = \left\{ {(x,y):|x| + |y| \le k} \right\}$$. If 27 (Area A1) = 5 (A...
The area (in sq. units) of the region enclosed between the parabola y2 = 2x and the line x + y = 4 is __________.
Let S be the region bounded by the curves y = x3 and y2 = x. The curve y = 2|x| divides S into two regions of areas R1, R2. If max {R1, R2} = R2, then...
If the line y = mx bisects the area enclosed by the lines x = 0, y = 0, x = $${3 \over 2}$$ and the curve y = 1 + 4x $$-$$ x2, then 12 m is equal to _...
Let a and b respectively be the points of local maximum and local minimum of the function f(x) = 2x3 $$-$$ 3x2 $$-$$ 12x. If A is the total area of th...
The area of the region $$S = \{ (x,y):3{x^2} \le 4y \le 6x + 24\}$$ is ____________.
The area (in sq. units) of the region bounded by the curves x2 + 2y $$-$$ 1 = 0, y2 + 4x $$-$$ 4 = 0 and y2 $$-$$ 4x $$-$$ 4 = 0, in the upper half pl...
Let T be the tangent to the ellipse E : x2 + 4y2 = 5 at the point P(1, 1). If the area of the region bounded by the tangent T, ellipse E, lines x = 1 ...
Let f : [$$-$$3, 1] $$\to$$ R be given as $$f(x) = \left\{ \matrix{ \min \,\{ (x + 6),{x^2}\}, - 3 \le x \le 0 \hfill \cr \max \,\{ \sqrt x ,{x... The area bounded by the lines y = || x$$-$$1 |$$-$$2 | is ___________. The graphs of sine and cosine functions, intersect each other at a number of points and between two consecutive points of intersection, the two graphs... ## MCQ (Single Correct Answer) The area of the region$$\left\{(x, y): x^{2} \leq y \leq\left|x^{2}-4\right|, y \geq 1\right\}$$is The area of the region enclosed by the curve$$f(x)=\max \{\sin x, \cos x\},-\pi \leq x \leq \pi$$and the$$x$$-axis is The area of the region enclosed by the curve$$y=x^{3}$$and its tangent at the point$$(-1,-1)$$is : Area of the region$$\left\{(x, y): x^{2}+(y-2)^{2} \leq 4, x^{2} \geq 2 y\right\}$$is The area of the region$$\left\{(x, y): x^{2} \leq y \leq 8-x^{2}, y \leq 7\right\}$$is : The area bounded by the curves$$y=|x-1|+|x-2|$$and$$y=3$$is equal to The area of the region given by$$\{ (x,y):xy \le 8,1 \le y \le {x^2}\} $$is : Let q be the maximum integral value of p in [0,10] for which the roots of the equation x^2-p x+\frac{5}{4} p=0 are rational. Then the area of ... The area of the region$$A = \left\{ {(x,y):\left| {\cos x - \sin x} \right| \le y \le \sin x,0 \le x \le {\pi \over 2}} \right\}$$is Let$$\Delta$$be the area of the region$$\left\{ {(x,y) \in {R^2}:{x^2} + {y^2} \le 21,{y^2} \le 4x,x \ge 1} \right\}$$. Then$${1 \over 2}\left( {\...
Let $$[x]$$ denote the greatest integer $$\le x$$. Consider the function $$f(x) = \max \left\{ {{x^2},1 + [x]} \right\}$$. Then the value of the integ...
Let $$A=\left\{(x, y) \in \mathbb{R}^{2}: y \geq 0,2 x \leq y \leq \sqrt{4-(x-1)^{2}}\right\}$$ and $$B=\left\{(x, y) \in \mathbb{R} \times \mathbb{R... The area enclosed by the curves$${y^2} + 4x = 4$$and$$y - 2x = 2$$is : The area of the region$$\left\{(x, y):|x-1| \leq y \leq \sqrt{5-x^{2}}\right\}$$is equal to : The area enclosed by the curves$$y=\log _{e}\left(x+\mathrm{e}^{2}\right), x=\log _{e}\left(\frac{2}{y}\right)$$and$$x=\log _{\mathrm{e}} 2$$, abov... The area of the region enclosed by$$y \leq 4 x^{2}, x^{2} \leq 9 y$$and$$y \leq 4$$, is equal to : Consider a curve$$y=y(x)$$in the first quadrant as shown in the figure. Let the area$$\mathrm{A}_{1}$$is twice the area$$\mathrm{A}_{2}$$. Then t... The area of the smaller region enclosed by the curves$$y^{2}=8 x+4$$and$$x^{2}+y^{2}+4 \sqrt{3} x-4=0$$is equal to The area bounded by the curves$$y=\left|x^{2}-1\right|$$and$$y=1$$is The odd natural number a, such that the area of the region bounded by y = 1, y = 3, x = 0, x = ya is$${{364} \over 3}$$, is equal to :... The area of the region given by$$A=\left\{(x, y): x^{2} \leq y \leq \min \{x+2,4-3 x\}\right\}$$is : Let the locus of the centre$$(\alpha, \beta), \beta>0$$, of the circle which touches the circle$$x^{2}+(y-1)^{2}=1$$externally and also touches the... The area enclosed by y2 = 8x and y =$$\sqrt2$$x that lies outside the triangle formed by y =$$\sqrt2$$x, x = 1, y = 2$$\sqrt2$$, is equal to:... The area of the bounded region enclosed by the curve$$y = 3 - \left| {x - {1 \over 2}} \right| - |x + 1|$$and the x-axis is : The area of the region S = {(x, y) : y2$$\le$$8x, y$$\ge\sqrt2$$x, x$$\ge$$1} is The area of the region bounded by y2 = 8x and y2 = 16(3$$-$$x) is equal to: The area bounded by the curve y = |x2$$-$$9| and the line y = 3 is : The area of the region enclosed between the parabolas y2 = 2x$$-$$1 and y2 = 4x$$-$$3 is The area, enclosed by the curves$$y = \sin x + \cos x$$and$$y = \left| {\cos x - \sin x} \right|$$and the lines$$x = 0,x = {\pi \over 2}$$, is : The area of the region bounded by the parabola (y$$-$$2)2 = (x$$-$$1), the tangent to it at the point whose ordinate is 3 and the x-axis is :... The area of the region bounded by y$$-$$x = 2 and x2 = y is equal to : If the area of the bounded region$$R = \left\{ {(x,y):\max \{ 0,{{\log }_e}x\} \le y \le {2^x},{1 \over 2} \le x \le 2} \right\}$$is ,$$\alpha {({...
The area (in sq. units) of the region, given by the set $$\{ (x,y) \in R \times R|x \ge 0,2{x^2} \le y \le 4 - 2x\}$$ is :
The area bounded by the curve 4y2 = x2(4 $$-$$ x)(x $$-$$ 2) is equal to :
Let A1 be the area of the region bounded by the curves y = sinx, y = cosx and y-axis in the first quadrant. Also, let A2 be the area of the region bou...
The area of the region : $$R = \{ (x,y):5{x^2} \le y \le 2{x^2} + 9\}$$ is :
The area (in sq. units) of the part of the circle x2 + y2 = 36, which is outside the parabola y2 = 9x, is :
The area (in sq. units) of the region enclosed by the curves y = x2 – 1 and y = 1 – x2 is equal to :
The area (in sq. units) of the region A = {(x, y) : |x| + |y| $$\le$$ 1, 2y2 $$\ge$$ |x|}
The area (in sq. units) of the region A = {(x, y) : (x – 1)[x] $$\le$$ y $$\le$$ 2$$\sqrt x$$, 0 $$\le$$ x $$\le$$ 2}, where [t] denotes the...
The area (in sq. units) of the region { (x, y) : 0 $$\le$$ y $$\le$$ x2 + 1, 0 $$\le$$ y $$\le$$ x + 1, $${1 \over 2}$$ $$\le$$ x $$\le$$...
Consider a region R = {(x, y) $$\in$$ R : x2 $$\le$$ y $$\le$$ 2x}. if a line y = $$\alpha$$ divides the area of region R into two equal parts,...
Area (in sq. units) of the region outside $${{\left| x \right|} \over 2} + {{\left| y \right|} \over 3} = 1$$ and inside the ellipse $${{{x^2}} \over ... Given :$$f(x) = \left\{ {\matrix{ {x\,\,\,\,\,,} & {0 \le x < {1 \over 2}} \cr {{1 \over 2}\,\,\,\,,} & {x = {1 \over 2}} \cr ...
The area (in sq. units) of the region {(x,y) $$\in$$ R2 : x2 $$\le$$ y $$\le$$ 3 – 2x}, is :
For a > 0, let the curves C1 : y2 = ax and C2 : x2 = ay intersect at origin O and a point P. Let the line x = b (0 < b < a) intersect the cho...
The area (in sq. units) of the region {(x, y) $$\in$$ R2 | 4x2 $$\le$$ y $$\le$$ 8x + 12} is :
The area of the region, enclosed by the circle x2 + y2 = 2 which is not common to the region bounded by the parabola y2 = x and the straight line y = ...
If the area (in sq. units) bounded by the parabola y2 = 4$$\lambda$$x and the line y = $$\lambda$$x, $$\lambda$$ > 0, is $${1 \over 9}$$ , then...
If the area (in sq. units) of the region {(x, y) : y2 $$\le$$ 4x, x + y $$\le$$ 1, x $$\ge$$ 0, y $$\ge$$ 0} is a $$\sqrt 2$$ + b, then a – b...
The area (in sq.units) of the region bounded by the curves y = 2x and y = |x + 1|, in the first quadrant is :
The area (in sq. units) of the region A = {(x, y) : $${{y{}^2} \over 2}$$ $$\le$$ x $$\le$$ y + 4} is :-
The area (in sq. units) of the region A = {(x, y) : x2 $$\le$$ y $$\le$$ x + 2} is
Let S($$\alpha$$) = {(x, y) : y2 $$\le$$ x, 0 $$\le$$ x $$\le$$ $$\alpha$$} and A($$\alpha$$) is area of the region S($$\alpha$$). If for a...
The area (in sq. units) of the region A = { (x, y) $$\in$$ R × R|  0 $$\le$$ x $$\le$$ 3, 0 $$\le$$ y $$\le$$ 4, y $$\le$$ x2 ...
The area (in sq. units) of the region bounded by the parabola, y = x2 + 2 and the lines, y = x + 1, x = 0 and x = 3, is
The area (in sq. units) in the first quadrant bounded by the parabola, y = x2 + 1, the tangent to it at the point (2, 5) and the coordinate axes is :...
The area (in sq. units) of the region bounded by the curve x2 = 4y and the straight line x = 4y – 2 is :
If the area enclosed between the curves y = kx2 and x = ky2, (k > 0), is 1 square unit. Then k is -
The area of the region A = {(x, y) : 0 $$\le$$ y $$\le$$x |x| + 1  and  $$-$$1 $$\le$$ x $$\le$$1} in sq. units, is : ...
The area (in sq. units) bounded by the parabolae y = x2 – 1, the tangent at the point (2, 3) to it and the y-axis is :
If the area of the region bounded by the curves, $$y = {x^2},y = {1 \over x}$$ and the lines y = 0 and x= t (t >1) is 1 sq. unit, then t is equal...
Let g(x) = cosx2, f(x) = $$\sqrt x$$ and $$\alpha ,\beta \left( {\alpha < \beta } \right)$$ be the roots of the quadratic equation 18x2 - 9$$\pi ... The area (in sq. units) of the region {x$$ \in $$R : x$$ \ge $$0, y$$ \ge $$0, y$$ \ge $$x$$-$$2 and y$$ \le \sqrt x $$}, is :... The area (in sq. units) of the smaller portion enclosed between the curves, x2 + y2 = 4 and y2 = 3x, is : The area (in sq. units) of the region$$\left\{ {\left( {x,y} \right):x \ge 0,x + y \le 3,{x^2} \le 4y\,and\,y \le 1 + \sqrt x } \right\}$$is The area (in sq. units) of the region described by A= {(x, y)$$\left| {} \right.$$y$$ \ge $$x2$$-$$5x + 4, x + y$$ \ge $$1, y$$ \le $$0} is : ... The area (in sq. units) of the region$$\left\{ {\left( {x,y} \right):{y^2} \ge 2x\,\,\,and\,\,\,{x^2} + {y^2} \le 4x,x \ge 0,y \ge 0} \right\}$$is : The area (in sq. units) of the region described by$$\left\{ {\left( {x,y} \right):{y^2} \le 2x} \right.$$and$$\left. {y \ge 4x - 1} \right\}$$is ... The area of the region described by$$A = \left\{ {\left( {x,y} \right):{x^2} + {y^2} \le 1} \right.$$and$$\left. {{y^2} \le 1 - x} \right\}$$is :... The area (in square units) bounded by the curves$$y = \sqrt {x,} 2y - x + 3 = 0,x$$-axis, and lying in the first quadrant is : The area between the parabolas$${x^2} = {y \over 4}$$and$${x^2} = 9y$$and the straight line$$y=2$$is : The area of the region enclosed by the curves$$y = x,x = e,y = {1 \over x}$$and the positive$$x$$-axis is : The area bounded by the curves$$y = \cos x$$and$$y = \sin x$$between the ordinates$$x=0$$and$$x = {{3\pi } \over 2}$$is The area of the region bounded by the parabola$${\left( {y - 2} \right)^2} = x - 1,$$the tangent of the parabola at the point$$(2, 3)$$and the$$x...
The area of the plane region bounded by the curves $$x + 2{y^2} = 0$$ and $$\,x + 3{y^2} = 1$$ is equal to :
The area enclosed between the curves $${y^2} = x$$ and $$y = \left| x \right|$$ is :
The area enclosed between the curve $$y = {\log _e}\left( {x + e} \right)$$ and the coordinate axes is :
The parabolas $${y^2} = 4x$$ and $${x^2} = 4y$$ divide the square region bounded by the lines $$x=4,$$ $$y=4$$ and the coordinate axes. If $${S_1},{S_... Let$$f(x)$$be a non - negative continuous function such that the area bounded by the curve$$y=f(x),x$$-axis and the ordinates$$x = {\pi \ov...
The area of the region bounded by the curves $$y = \left| {x - 2} \right|,x = 1,x = 3$$ and the $$x$$-axis is :
The area of the region bounded by the curves $$y = \left| {x - 1} \right|$$ and $$y = 3 - \left| x \right|$$ is :
The area bounded by the curves $$y = \ln x,y = \ln \left| x \right|,y = \left| {\ln {\mkern 1mu} x} \right|$$ and y = \left| {\ln \left| x \right|} ...
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