Application of Integration

Consider the function $y=f(x)$ defined implicitly by the equation $y^3-3 y+x=0$ on the interval $(-\infty,-2) \cup(2, \infty)$. The area of the region bounded by the curve $y=f(x)$, the $x$-axis and the lines $x=a, x=b$, where $-\infty< a< ...

Consider the function \(\mathrm{f}(x)=(x-2) \log _{\mathrm{e}} x\). Then the equation \(x \log _{\mathrm{e}} x=2-x\)

The area bounded by the curves \(x=4-y^2\) and the Y-axis is

Let f be a non-negative function defined in \([0,\pi /2]\), f' exists and be continuous for all x and \(\int\limits_0^x {\sqrt {1 - {{(f'(t))}^2}} dt = \int\limits_0^x {f(t)dt} }\) and f (0) = 0. Then

Area of the figure bounded by the parabola \({y^2} + 8x = 16\) and \({y^2} - 24x = 48\) is

The area bounded by the parabolas \(y = 4{x^2},y = {{{x^2}} \over 9}\) and the straight line y = 2 is

The straight the through the origin which divides the area formed by the curves y = 2x \(-\) x2, y = 0 and x = 1 into two equal halves is

If \({x^2} + {y^2} = {a^2}\), then \(\int\limits_0^a {\sqrt {1 + {{\left( {{{dy} \over {dx}}} \right)}^2}} dx = }\)

The area of the region\(\{ (x,y):{x^2} + {y^2} \le 1 \le x + y\}\) is

The area of the figure bounded by the parabola \(x = - 2{y^2},\,x = 1 - 3{y^2}\) is

Area in the first quadrant between the ellipses x2 + 2y2 = a2 and 2x2 + y2 = a2 is

Let P and T be the subsets of k, y-plane defined byP = {(x, y) : x > 0, y > 0 and x2 + y2 = 1}T = {(x, y) : x > 0, y > 0 and x8 + y8 < 1}Then, P \(\cap\) T is

The area bounded by y = x + 1 and y = cos x and the X-axis, is

The area of the figure bounded by the parabolas x = \(-\) 2y2 and x = 1 \(-\) 3y2 is

The area bounded by y2 = 4x and x2 = 4y is

The area enclosed between y2 = x and y = x is

The area of the region bounded by y2 = x and y = | x | is

The area bounded by the parabolas y = 4x2, \(y = {{{x^2}} \over 9}\) and the line y = 2 is

The area enclosed by y = 3x \(-\) 5, y = 0, x = 3 and x = 5 is

The area included between the parabolas y2 = 4x and x2 = 4y is

The line which is parallel to x-axis and crosses the curve y = \(\sqrt x\) at an angle 45\(^\circ\) is

The area enclosed between the curve y = 1 + x2, the y-axis, and the straight line y = 5 is given by