Application of Integration · Mathematics · JEE Advanced

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JEE Advanced 2024 Paper 2 Online
Let $S=\left\{(x, y) \in \mathbb{R} \times \mathbb{R}: x \geq 0, y \geq 0, y^2 \leq 4 x, y^2 \leq 12-2 x\right.$ and $\left.3 y+\sqrt{8} x \leq 5 \sqr... JEE Advanced 2021 Paper 1 Online The area of the region $$\left\{ {\matrix{ {(x,y):0 \le x \le {9 \over 4},} & {0 \le y \le 1,} & {x \ge 3y,} & {x + y \ge 2} \cr } ... JEE Advanced 2020 Paper 1 Offline Let the functions f : R$$ \to $$R and g : R$$ \to $$R be defined byf(x) = ex$$-$$1$$-$$e$$-$$|x$$-$$1|and g(x) =$${1 \over 2}$$(ex$$-$$1 ... JEE Advanced 2019 Paper 1 Offline The area of the region{(x, y) : xy$$ \le $$8, 1$$ \le $$y$$ \le $$x2} is JEE Advanced 2016 Paper 2 Offline Area of the region$$\left\{ {\left( {x,y} \right) \in {R^2}:y \ge \sqrt {\left| {x + 3} \right|} ,5y \le x + 9 \le 15} \right\}$$is equal to ... JEE Advanced 2013 Paper 1 Offline The area enclosed by the curves$$y = \sin x + {\mathop{\rm cosx}\nolimits} $$and$$y = \left| {\cos x - \sin x} \right|$$over the interval$$\left[... IIT-JEE 2011 Paper 1 Offline Let the straight line $$x=b$$ divide the area enclosed by $$y = {\left( {1 - x} \right)^2},y = 0,$$ and $$x=0$$ into two parts $${R_1}\left( {0 \le x... IIT-JEE 2011 Paper 2 Offline Let f$$:\left[ { - 1,2} \right] \to \left[ {0,\infty } \right]$$be a continuous function such that$$f\left( x \right) = f\left( {1 - x} \right... IIT-JEE 2010 Paper 2 Offline Consider the polynomial $$f\left( x \right) = 1 + 2x + 3{x^2} + 4{x^3}.$$ Let $$s$$ be the sum of all distinct real roots of $$f(x)$$ and let $$t = \l... IIT-JEE 2009 Paper 1 Offline Let$$f$$be a non-negative function defined on the interval$$[0,1]$$. If$$\int\limits_0^x {\sqrt {1 - {{(f'(t))}^2}dt} = \int\limits_0^x {f(t)dt,... IIT-JEE 2008 Paper 2 Offline The area of the region between the curves $$y = \sqrt {{{1 + \sin x} \over {\cos x}}}$$ and $$y = \sqrt {{{1 - \sin x} \over {\cos x}}}$$ bounded b... IIT-JEE 2008 Paper 1 Offline The area of the region bounded by the curve $$y=f(x),$$ the $$x$$-axis, and the lines $$x=a$$ and $$x=b$$, where $$- \infty < a < b < - 2,... IIT-JEE 2005 Screening The area bounded by the parabola$$y = {\left( {x + 1} \right)^2}$$and$$y = {\left( {x - 1} \right)^2}$$and the line$$y=1/4$$is IIT-JEE 2004 Screening The area enclosed between the curves$$y = a{x^2}$$and$$x = a{y^2}\left( {a > 0} \right)$$is$$1$$sq. unit, then the value of$$a$$is IIT-JEE 2003 Screening The area bounded by the curves$$y = \sqrt x ,2y + 3 = x$$and$$x$$-axis in the 1st quadrant is IIT-JEE 2002 Screening The area bounded by the curves$$y = \left| x \right| - 1$$and$$y = - \left| x \right| + 1$$is IIT-JEE 2002 Screening Let$$f\left( x \right) = \int\limits_1^x {\sqrt {2 - {t^2}} \,dt.} $$Then the real roots of the equation$${x^2} - f'\left( x \right) = 0$$are ... IIT-JEE 1997 If$$g\left( x \right) = \int_0^x {{{\cos }^4}t\,dt,} $$then$$g\left( {x + \pi } \right)$$equals IIT-JEE 1982 The area bounded by the curves$$y=f(x)$$, the$$x$$-axis and the ordinates$$x=1$$and$$x=b$$is$$(b-1)$$sin$$(3b+4)$$. Then$$f(x)$$is Numerical JEE Advanced 2024 Paper 2 Online Let the function f:[1, \infty) \rightarrow \mathbb{R} be defined by$$ f(t)=\left\{\begin{array}{cc} (-1)^{n+1} 2, & \text { if } t=2 n-1, n \in \m... JEE Advanced 2023 Paper 1 Online Let$n \geq 2$be a natural number and$f:[0,1] \rightarrow \mathbb{R}$be the function defined by $$f(x)= \begin{cases}n(1-2 n x) & \text { if } 0 \... JEE Advanced 2022 Paper 2 Online Consider the functions f, g: \mathbb{R} \rightarrow \mathbb{R} defined by$$ f(x)=x^{2}+\frac{5}{12} \quad \text { and } \quad g(x)= \begin{cases}2... JEE Advanced 2021 Paper 2 Online Let f1 : (0, $$\infty$$) $$\to$$ R and f2 : (0, $$\infty$$) $$\to$$ R be defined by $${f_1}(x) = \int\limits_0^x {\prod\limits_{j = 1}^{21} {{{(t - j)... JEE Advanced 2021 Paper 2 Online Let f1 : (0,$$\infty$$)$$\to$$R and f2 : (0,$$\infty$$)$$\to$$R be defined by$${f_1}(x) = \int\limits_0^x {\prod\limits_{j = 1}^{21} {{{(t - j)... JEE Advanced 2018 Paper 1 Offline A farmer F1 has a land in the shape of a triangle with vertices at P(0, 0), Q(1, 1) and R(2, 0). From this land, a neighbouring farmer F2 takes away t... JEE Advanced 2015 Paper 2 Offline Let $$f:R \to R$$ be a continuous odd function, which vanishes exactly at one point and $$f\left( 1 \right) = {1 \over {2.}}$$ Suppose that $$F\left( ... JEE Advanced 2015 Paper 1 Offline Let$$F\left( x \right) = \int\limits_x^{{x^2} + {\pi \over 6}} {2{{\cos }^2}t\left( {dt} \right)} $$for all$$x \in R$$and$$f:\left[ {0,{1 \over ... MCQ (More than One Correct Answer) JEE Advanced 2021 Paper 2 Online For any real numbers $$\alpha$$ and $$\beta$$, let $${y_{\alpha ,\beta }}(x)$$, x$$\in$$R, be the solution of the differential equation $${{dy} \over ... JEE Advanced 2018 Paper 1 Offline Let f : [0,$$\infty $$)$$ \to $$R be a continuous function such that$$f(x) = 1 - 2x + \int_0^x {{e^{x - t}}f(t)dt} $$for all x$$ \in $$[0,$$\in... JEE Advanced 2017 Paper 2 Offline If the line x = $$\alpha$$ divides the area of region R = {(x, y) $$\in$$R2 : x3 $$\le$$ y $$\le$$ x, 0 $$\le$$ x $$\le$$ 1} into two equal... JEE Advanced 2015 Paper 2 Offline Let $$F:R \to R$$ be a thrice differentiable function. Suppose that $$F\left( 1 \right) = 0,F\left( 3 \right) = - 4$$ and $$F\left( x \right) < 0... IIT-JEE 2012 Paper 1 Offline Let$$S$$be the area of the region enclosed by$$y = {e^{ - {x^2}}}$$,$$y=0$$,$$x=0$$, and$$x=1$$; then IIT-JEE 2010 Paper 1 Offline Let$$f$$be a real-valued function defined on the interval$$\left( {0,\infty } \right)$$by$$\,f\left( x \right) = \ln x + \int\limits_0^x {\sqrt ... IIT-JEE 2009 Paper 1 Offline Area of the region bounded by the curve $$y = {e^x}$$ and lines $$x=0$$ and $$y=e$$ is IIT-JEE 1999 For which of the following values of $$m$$, is the area of the region bounded by the curve $$y = x - {x^2}$$ and the line $$y=mx$$ equals $$9/2$$? Subjective IIT-JEE 2006 Match the following : Column $$I$$ (A) $$\int\limits_0^{\pi /2} {{{\left( {\sin x} \right)}^{\cos x}}\left( {\cos x\cot x - \log {{\left( {\sin x} \ri... IIT-JEE 2005 Find the area bounded by the curves$${x^2} = y,{x^2} = - y$$and$${y^2} = 4x - 3.$$IIT-JEE 2005 If$$\left[ {\matrix{ {4{a^2}} & {4a} & 1 \cr {4{b^2}} & {4b} & 1 \cr {4{c^2}} & {4c} & 1 \cr } } \right]\le... IIT-JEE 2002 Find the area of the region bounded by the curves $$y = {x^2},y = \left| {2 - {x^2}} \right|$$ and $$y=2,$$ which lies to the right of the line $$x=1.... IIT-JEE 2001 Let$$b \ne 0$$and for$$j=0, 1, 2, ..., n,$$let$${S_j}$$be the area of the region bounded by the$$y$$-axis and the curve$$x{e^{ay}} = \sin $$... IIT-JEE 1999 Let$$f(x)$$be a continuous function given by$$$f\left( x \right) = \left\{ {\matrix{ {2x,} & {\left| x \right| \le 1} \cr {{x^2} + ax ...
IIT-JEE 1997
Let $$f(x)= Maximum$$ $$\,\left\{ {{x^2},{{\left( {1 - x} \right)}^2},2x\left( {1 - x} \right)} \right\},$$ where $$0 \le x \le 1.$$ Determine the ...
IIT-JEE 1996
Let $${A_n}$$ be the area bounded by the curve $$y = {\left( {\tan x} \right)^n}$$ and the lines $$x=0,$$ $$y=0,$$ and $$x = {\pi \over 4}.$$ Prove ...
IIT-JEE 1995
Consider a square with vertices at $$(1,1), (-1,1), (-1,-1)$$ and $$(1, -1)$$. Let $$S$$ be the region consisting of all points inside the square whic...
IIT-JEE 1994
In what ratio does the $$x$$-axis divide the area of the region bounded by the parabolas $$y = 4x - {x^2}$$ and $$y = {x^2} - x?$$
IIT-JEE 1992
Sketch the region bounded by the curves $$y = {x^2}$$ and $$y = {2 \over {1 + {x^2}}}.$$ Find the area.
IIT-JEE 1991
Sketch the curves and identify the region bounded by $$x = {1 \over 2},x = 2,y = \ln \,x$$ and $$y = {2^x}.$$ Find the area of this region.
IIT-JEE 1991
If $$'f$$ is a continuous function with $$\int\limits_0^x {f\left( t \right)dt \to \infty }$$ as $$\left| x \right| \to \infty ,$$ then show that eve...
IIT-JEE 1990
Compute the area of the region bounded by the curves $$\,y = ex\,\ln x$$ and $$y = {{\ln x} \over {ex}}$$ where $$ln$$ $$e=1.$$
IIT-JEE 1988
Find the area of the region bounded by the curve $$C:y=$$ $$\tan x,$$ tangent drawn to $$C$$ at $$x = {\pi \over 4}$$ and the $$x$$-axis.
IIT-JEE 1987
Find the area bounded by the curves, $${x^2} + {y^2} = 25,\,4y = \left| {4 - {x^2}} \right|$$ and $$x=0$$ above the $$x$$-axis.
IIT-JEE 1985
Sketch the region bounded by the curves $$y = \sqrt {5 - {x^2}}$$ and $$y = \left| {x - 1} \right|$$ and find its area.
IIT-JEE 1984
Find the area of the region bounded by the $$x$$-axis and the curves defined by $$y = \tan x, - {\pi \over 3} \le x \le {\pi \over 3};\,\,y = \cot... IIT-JEE 1983 Find the area bounded by the$$x$$-axis, part of the curve$$y = \left( {1 + {8 \over {{x^2}}}} \right)$$and the ordinates at$$x=2$$and$$x=4$$. I... IIT-JEE 1982 For any real$$t,\,x = {{{e^t} + {e^{ - t}}} \over 2},\,\,y = {{{e^t} - {e^{ - t}}} \over 2}$$is a point on the hyperbola$${x^2} - {y^2} = 1$$. Sho... IIT-JEE 1981 Find the area bounded by the curve$${x^2} = 4y and the straight
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