Embibe Experts Solutions for Chapter: Applications of Integrals, Exercise 1: Exercise

Show that the area of the region bounded by $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$ is $\mathrm{\pi }ab$. Also deduce area of the circle $x^2+y^2=a^2$.

Using integration, find the area of the region bounded by the curves $y=\sqrt{5-x^2}$ and $y=\left|x-1\right|$.

Find the area of the region enclosed by the curves $y=\sin 2x, y=\sqrt{3}\sin x, x=0, x=\frac{\mathrm{\pi }}{6}$.

Show that the area enclosed between the curves $y^2=12\left(x+3\right)$ and $y^2=20\left(5-x\right)$ is $64\sqrt{\frac{5}{3}}$.

The circle $x^2+y^2=8$ and is divided into parts by parabola $2y=x^2$. Find the area of both the parts.

Find the area of the region bounded by the curves $y=\mathrm{sin\pi }x, y=x^2-x, x=2$.

Let $AOB$ be the positive quadrant of the ellipse $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$ with $OA=a, OB=b$. Then show that the area bounded between the chord $AB$ and arc $AB$ of the ellipse is $\frac{ab\left(\mathrm{\pi }-2\right)}{4}$.

Prove that the curves $y^2=4x$ and $x^2=4y$ divide the area of the square bounded by the lines $x=0, x=4, y=4, y=0$ into three equal parts.