Application of Integration

Consider the ellipses given by
\(x^2+4 y^2=1 \quad \text { and } \quad 4 x^2+y^2=1 .\)
If $\alpha$ is the area of the common region that lies inside both the given ellipses, then the value of $\cot \alpha$ is $\_\_\_\_$ .

Consider the curve $C_1$ given by
\(y=e^{-x} \quad \text { for } x \in[0,10 \pi],\)
and the curve $C_2$ given by
\(y=e^{-x}(\sin x+\cos x) \quad \text { for } x \in[0,10 \pi] .\)
Let $n$ be the total number of points of intersection of ...

Let ℝ denote the set of all real numbers. Then the area of the region
$ \left\{ (x, y) \in \mathbb{R} \times \mathbb{R} : x > 0, y > \frac{1}{x}, 5x - 4y - 1 > 0, 4x + 4y - 17 < 0 \right\} $
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 \mathbb{N}, \\ \frac{(2 n+1-t)}{2} f(2 n-1)+\frac{(t-(2 n-1))}{2} f(2 n+1), & \text { if } 2...

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 $\left.3 y+\sqrt{8} x \leq 5 \sqrt{8}\right\}$. If the area of the region $S$ is $\alpha \sqrt{2}$, then $\alpha$ is equal ...

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 \leq x \leq \frac{1}{2 n} \\\\ 2 n(2 n x-1) & \text { if } \frac{1}{2 n} \leq x \leq \frac{...

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\left(1-\frac{4|x|}{3}\right), & |x| \leq \frac{3}{4} \\ 0, & |x|>\frac{3}{4}\end{cases}
$...

Let f1 : (0, \(\infty\)) \(\to\) R and f2 : (0, \(\infty\)) \(\to\) R be defined by \({f_1}(x) = \int\limits_0^x {\prod\limits_{j = 1}^{21} {{{(t - j)}^j}dt} }\), x > 0 and \({f_2}(x) = 98{(x - 1)^{50}} - 600{(x - 1)^{49}} + 2450,x > 0\), ...

Let f1 : (0, \(\infty\)) \(\to\) R and f2 : (0, \(\infty\)) \(\to\) R be defined by \({f_1}(x) = \int\limits_0^x {\prod\limits_{j = 1}^{21} {{{(t - j)}^j}dt} }\), x > 0 and \({f_2}(x) = 98{(x - 1)^{50}} - 600{(x - 1)^{49}} + 2450,x > 0\), ...

For any real numbers \(\alpha\) and \(\beta\), let \({y_{\alpha ,\beta }}(x)\), x\(\in\)R, be the solution of the differential equation \({{dy} \over {dx}} + \alpha y = x{e^{\beta x}},y(1) = 1\). Let $$S = \{ {y_{\alpha ,\beta }}(x):\alpha ...

The area of the region \(\left\{ {\matrix{ {(x,y):0 \le x \le {9 \over 4},} & {0 \le y \le 1,} & {x \ge 3y,} & {x + y \ge 2} \cr } } \right\}\) is

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 + e1 \(-\) x).The the area of the region in the first quadrant bounded by the curves y = f...

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

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, \(\infty\)). Then, which of the following statement(s) is (are) TRUE?

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 the region which lies between the sides PQ and a curve of the form y = xn (n > 1). If the a...

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 parts, then

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

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 \(F\left( x \right) = \int\limits_{ - 1}^x {f\left( t \right)dt}\) for all $$x \in \,\,\left[ { - 1...

Let \(F:R \to R\) be a thrice differentiable function. Suppose that
\(F\left( 1 \right) = 0,F\left( 3 \right) = - 4\) and \(F\left( x \right) < 0\) for all \(x \in \left( {{1 \over 2},3} \right).\) Let $$f\left( x \right) = xF\left( x \ri...

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 \(f:\left[ {0,{1 \over 2}} \right] \to \left[ {0,\infty } \right]\) be a continuous function. For $$a \in \left[ ...

The area enclosed by the curves \(y = \sin x + {\mathop{\rm cosx}\nolimits}\) and \(y = \left| {\cos x - \sin x} \right|\) over the interval \(\left[ {0,{\pi \over 2}} \right]\) is

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

Let f \(:\)\(\left[ { - 1,2} \right] \to \left[ {0,\infty } \right]\) be a continuous function such that
\(f\left( x \right) = f\left( {1 - x} \right)\) for all \(x \in \left[ { - 1,2} \right]\)
Let $${R_1} = \int\limits_{ - 1}^2 {xf\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 \({R_1}\left( {0 \le x \le b} \right)\) and
\({R_2}\left( {b \le x \le 1} \right)\) such that $${R_1} - {R_2} =...

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 \(t = \left| s \right|.\)
The area bounded by the curve \(y=f(x)\) and the lines \(x=0,\) \(y=0\) ...

Let \(f\) be a real-valued function defined on the interval \(\left( {0,\infty } \right)\)
by \(\,f\left( x \right) = \ln x + \int\limits_0^x {\sqrt {1 + \sin t\,} dt.}\) then which of the following
statement(s) is (are) true?

Let \(f\) be a non-negative function defined on the interval \([0,1]\).
If \(\int\limits_0^x {\sqrt {1 - {{(f'(t))}^2}dt} = \int\limits_0^x {f(t)dt,0 \le x \le 1} }\), and \(f(0) = 0\), then

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

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 by the lines \(x=0\) and \(x = {\pi \over 4}\) is

The area of the region bounded by the curve \(y=f(x),\) the
\(x\)-axis, and the lines \(x=a\) and \(x=b\), where \(- \infty < a < b < - 2,\) is :

\(\text { Match the following : }\)


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\(f''(x) < 0 \forall x \in(a, b)\) and \(c\) is a point such that \(a < c < b\), and \((c, f(C))\) is the point lying on the curve for which \(\mathrm{F}(C)\) is maximum, then \(f'(C)\) is equal to:

If \(\lim_\limits{t \rightarrow a} \frac{\int_{a}^{t} f(x) d x-\frac{(t-a)}{2}\{f(t)+f(a)\}}{(t-a)^{3}}=0\) then the degree of polynomial function \(f(x)\) almost is:

\(\int_\limits{0}^{\pi / 2} \sin x d x\) is equal to:

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

If $$\left[\begin{array}{lll}4 a^{2} & 4 a & 1 \\ 4 b^{2} & 4 b & 1 \\ 4 c^{2} & 4 c & 1\end{array}\right]\left[\begin{array}{c}f(-1) \\ f(1) \\ f(2)\end{array}\right]=\left[\begin{array}{c}3 a^{2}+3 a \\ 3 b^{2}+3 b \\ 3 c^{2}+3 c\end{arra...

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

If length of tangent at any point on the curve
\(y = f(x)\) intercepted between the point and
the X-axis is of length 1. Find the equation of
the curve.

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

} } \right]\left[ {\matrix{
{f\left( { - 1} \right)} \cr
{f\left( 1 \right)} \cr
{f\left( 2 \right)} \cr

} } \r...

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

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

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 \(x{e^{ay}} = \sin\) by,
\({{jr} \over b} \le y \le {{\left( {j + 1} \right)\pi } \over b}.\) Show that $${S_...

Let \(f(x)\) be a continuous function given by
\($f\left( x \right) = \left\{ {\matrix{ {2x,} & {\left| x \right| \le 1} \cr {{x^2} + ax + b,} & {\left| x \right| > 1} \cr } } \right\}\)$
Find the area of the region in the thir...

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 \(9/2\)?

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

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 area of the region bounded by the curves
\(y = f\left( x \right),\) \(x\)-axes, \(x=0\) a...