Application of Integration

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 that for \(n > 2,\)
\({A_n} + {A_{n - 2}} = {1 \over {n - 1}}\) and deduce $${1 \over {2n ...

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 which are nearer to the origin than to any edge. Sketch the region \(S\) and find its area.

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.

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 every line \(y=mx\)

intersects the curve $${y^2} + \int\limits_0^x {f\left( t \right)dt = ...

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.

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
\($y = \tan x, - {\pi \over 3} \le x \le {\pi \over 3};\,\,y = \cot x,{\pi \over 6} \le x \le {{3\pi } \over 2}\)$

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 \(x=4\). If the ordinate at \(x=a\) divides the area into two equal parts, find \(a\).

For any real \(t,\,x = {{{e^t} + {e^{ - t}}} \over 2},\,\,y = {{{e^t} - {e^{ - t}}} \over 2}\) is a point on the
hyperbola \({x^2} - {y^2} = 1\). Show that the area bounded by this hyperbola and the lines joining its centre to the points c...

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

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