Application of Integration · Mathematics · JEE Advanced

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...

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 } ...

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 ...

The area of the region{(x, y) : xy $$ \le $$ 8, 1 $$ \le $$ y $$ \le $$ x2} is

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 ...

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[...

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...

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...

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...

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,...

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...

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,...

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

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

The area bounded by the curves $$y = \sqrt x ,2y + 3 = x$$ and $$x$$-axis in the 1st quadrant is

The area bounded by the curves $$y = \left| x \right| - 1$$ and $$y = - \left| x \right| + 1$$ is

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 ...

If $$g\left( x \right) = \int_0^x {{{\cos }^4}t\,dt,} $$ then $$g\left( {x + \pi } \right)$$ equals

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

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...

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 \...

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...

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)...

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)...

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...

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( ...

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 ...

For any real numbers $$\alpha$$ and $$\beta$$, let $${y_{\alpha ,\beta }}(x)$$, x$$\in$$R, be the solution of the differential equation $${{dy} \over ...

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...

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...

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...

Let $$S$$ be the area of the region enclosed by $$y = {e^{ - {x^2}}}$$, $$y=0$$, $$x=0$$, and $$x=1$$; then

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 ...

Area of the region bounded by the curve $$y = {e^x}$$ and lines $$x=0$$ and $$y=e$$ is

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 $$9/2$$?

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...

If $$\left[ {\matrix{ {4{a^2}} & {4a} & 1 \cr {4{b^2}} & {4b} & 1 \cr {4{c^2}} & {4c} & 1 \cr } } \right]\le...

Find the area bounded by the curves $${x^2} = y,{x^2} = - y$$ and $${y^2} = 4x - 3.$$

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....

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 $$x{e^{ay}} = \sin $$...

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 ...

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 ...

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 ...

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...

In what ratio does the $$x$$-axis divide the area of the region bounded by the parabolas $$y = 4x - {x^2}$$ and $$y = {x^2} - x?$$

Sketch the region bounded by the curves $$y = {x^2}$$ and $$y = {2 \over {1 + {x^2}}}.$$ Find the area.

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.

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...

Compute the area of the region bounded by the curves $$\,y = ex\,\ln x$$ and $$y = {{\ln x} \over {ex}}$$ where $$ln$$ $$e=1.$$

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.

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.

Sketch the region bounded by the curves $$y = \sqrt {5 - {x^2}} $$ and $$y = \left| {x - 1} \right|$$ and find its area.

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...

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...

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...

Find the area bounded by the curve $${x^2} = 4y$$ and the straight