Embibe Experts Solutions for Chapter: Area under Curves, Exercise 2: EXERCISE-2

The value of $\text{'}a\text{'}$ $(a>0)$ for which the area bounded by the curves $y=\frac{x}{6}+\frac{1}{x^2}, y=0,$$x=a$ and $x=2a$ has the least value, is -

Consider the following regions in the plane:

$R_1=\left\{\left(x, y\right) : 0\le x\le 1 \& 0\le y\le 1\right\}$ and $R_2=\left\{\left(x, y\right) : x^2+y^2\le 4/3\right\}$.

The area of the region $R_1\cap R_2$ can be expressed as $\frac{a\sqrt{3}+b\pi }{9}$, where $a$ and $b$ are integers, then - 

The area of the region of the plane bounded by $\left(\left|x\right|,\left|y\right|\right)\le 1$ and $xy\le \frac{1}{2}$ is

The line $y=mx$ bisects the area enclosed by the curve $y=1+4x-x^2$& the lines $x=0,$ $x=\frac{3}{2}$ $\&$ $y=0.$ Then the value of $m$ is

 Area of the region enclosed between the curves  $x=y^2-1$  and  $x=\left|y\right|\sqrt{1-y^2}$ is 

If the tangent to the curve $y=1-x^2$ at $x=\alpha$, where $0<\alpha <1$, meets the axes at $P$ and $Q.$ As $\alpha$ varies, the minimum value of the area of the triangle $OPQ$ is $k$ times the area bounded by the axes and the part of the curve for which $0<x<1$, then $k$ is equal to -

Let $\text{'}a\text{'}$ be a positive constant. Consider two curves $C_1: y=e^x, C_2: y=e^{a-x}.$ Let $S$ be the area of the part surrounding by $C_1, C_2$ and the $y$-axis, then -

If $(a, 0) \& (b, 0), [a, b>0]$ are the points where the curve $y=\sin 2x-\sqrt{3}\sin x$ cuts the positive $x$-axis first $\&$ second time, $A \& B$ are the areas bounded by the curve $\&$ positive $x$-axis between $x=0$ to $x=a$ and $x=a$ to $x=b$ respectively, then