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

Obtain the area enclosed by region bounded by the curves $y=x\ln x$ and $y=2x-2x^2$.

Find the value of $m\left(m>0\right)$ for which the area bounded by the line $y=mx+2$ and $x=2y-y^2$ is $\frac{9}{2}$ square units.

If the area bounded by $y=f^{-1}\left(x\right), x=10, x=4$ and $x$-axis given that area bounded by $y=f\left(x\right), x=2, x=6$ and $x$-axis is $30 \mathrm{sq}.\mathrm{units}$, where $f\left(2\right)=4$ and $f\left(6\right)=10$ is $k$ sq. units, then the value of $k$ is,

 (given $f\left(x\right)$ is an invertible function)

Consider a line $\mathrm{\ell } : 2x-\sqrt{3}y=0$ and a parametrised $C : x=\tan t, y=\frac{1}{\cos t}\left(0\le t<\frac{\pi }{2}\right)$. If the area of part bounded by $\mathrm{\ell }, C$ and the $y$-axis is equal to $\frac{1}{4}\ln \left(a+\sqrt{b}\right)$, where $a, b, \in N, b$, is not perfect square then find the value of $\left(a+b\right)$.

Area bounded by $y={\sin }^{-1}x, y={\cos }^{-1}x, y=0$ in the first quadrant is equal to

Let $f\left(x\right)$ be a non-negative, continuous and even function such that area bounded by $x$-axis, $y$-axis & $y=f\left(x\right)$ is equal to $\left(x^2+x^3\right)$ square units $\forall x\ge 0,$ then

Let $\text{'}c\text{'}$ be a positive real number such that area bounded by $y=0,y=\left[{\tan }^{-1}x\right]$ from $x=0$ to $x=c$ is equal to area bounded by $y=0, y=\left[{\cot }^{-1}x\right]$ from $x=0$ to $x=c$ (where $\left[*\right]$ represents greatest integer function), then

Area bounded by $y=x^2-2\left|x\right|$ and $y=-1$ is equal to