Embibe Experts Solutions for Chapter: Area under Curves, Exercise 3: Exercise-3

Area of the region bounded by the curve $y=e^x$ and lines $x=0$ and $y=e$ is 

Let the straight line $x=b$ divide the area enclosed by $y={\left(1-x\right)}^2, y=0$ and $x=0$ into two parts $R_1\left(0\le x\le b\right)$ and $R_2\left(b\le x\le 1\right)$ such that $R_1-R_2=\frac{1}{4}$. Then $b$ equals

Let $f: \left[-1,2\right]\to [0,\mathit{\infty })$ be a continuous function such that $f\left(x\right)=f\left(1-x\right) \forall x\in \left[-1,2\right]$. Let $R_1={\int }_{-1}^{2}x f\left(x\right) dx$ and $R_2$ be the area of the region bounded by $y=f\left(x\right), x=-1, x=2$ and the $X-$axis. Then,

A farmer $F_1$ has a land in the shape of a triangle with vertices at $P(0,0), Q(1,1)$ and $R(2,0)$. From this land, a neighbouring farmer $F_2$ takes away the region which lies between the side $PQ$ and a curve of the form $y=x^n(n>1)$. If the area of the region taken away by the farmer $F_2$ is exactly $30\%$ of the area of $∆PQR$, then the value of $n$ is

The area bounded by the curves $y=\cos x$ and $y=\sin x$ between the ordinates $x=0$ and $x=\frac{3\pi }{2}$ is

The area of the region enclosed by the curves $y=x, x=e, y=\frac{1}{x}$ and the positive $x$-axis is

The area $\left(\mathrm{in} \mathrm{sq}. \mathrm{units}\right)$ bounded by the curves $y^2=4x\text{ and }{x}^2=4y$ is 

The area in $\left(\mathrm{sq}. \mathrm{units}\right)$ bounded between the parabolas $x^2=\frac{y}{4}$ and $x^2=9y,$ and the straight line $y=2$ is