$\text{Given }f\left(x\right)=\stackrel{x}{\underset{0}{\int }}{e}^t\left({\log }_e\sec t-{\sec }^{2}t\right)dt,g\left(x\right)=-2e^x\tan x,$ then the area bounded by the curves $y=f\left(x\right)\text{and}y=g\left(x\right)$ between the ordinates $x=0\text{and}x=\frac{\pi }{3},$ is (in sq. units)

The area enclosed by $y=x^2+\cos x\text{ and its normal at }x=\frac{\pi }{2}$ in the first quadrant is

Which of the following have the same bounded area

Let A(k) be the area bounded by the curves $y=x^2-3$ and y=kx+2\displaystyle.

The area of the region containing the points (x,y) satisfy- ing $4\le x^2+y^2\le 2\left(\left|x\right|+\left|y\right|\right)$ is

The area enclosed between the curve $y^2\left(2a-x\right)=x^3$ and the line x=2 above the x-axis is

The parabolas $y^2=4x\text{and}{x}^2=4y$ divide the square region bounded by the lines x=4, y=4 and the coordinate axes. If $S_1,S_2,S_3$ are the areas of these parts numbered from top to bottom, respectively, then

The area bounded by the two branches of curve ${\left(y-x\right)}^2=x^3$ and the straight line x=1 is

Let f(x) be a non-negative continuous function such that the area bounded by the curve y=f(x), the x-axis, and the ordinates $x=\frac{\pi }{4}\text{and}x=\beta >\frac{\pi }{4}\text{ is }\beta \sin \beta +\frac{\pi }{4}\cos \beta +\sqrt{2}\beta .$ Then $f^{\prime }\left(\frac{\pi }{2}\right)$ is

The area bounded by the curves $y={\sin }^{-1}\left|\sin x\right|\text{and}y={\left({\sin }^{-1}\left|\sin x\right|\right)}^2,\text{ where }0\le x\le 2\pi$, is