Name: Area Between Curve and X-axis
Uploaded: 2019-07-19
Duration: 1 min 50 s
Description: As you might have seen, definite integration is used to find the area bounded by a curve. In this video, we will learn and understand the area bounded by a c...

Question

Determine the area within the ellipse

\frac{x^{2}}{a^{2}} + \frac{y^{2}}{b^{2}} = 1

.

Accepted Answer

We have to find area enclosed by ellipse.

Since Ellipse is symmetrical about both $x -$ axis and $y -$ axis.

$∴$ Area of ellipse = $4 \times$ Area of $O A B$ $= 4 \times \int_{0}^{a} y d x$

We know that, $\frac{x^{2}}{a^{2}} + \frac{y^{2}}{b^{2}} = 1$

$\Rightarrow \frac{y^{2}}{b^{2}} = 1 - \frac{x^{2}}{a^{2}} = \frac{a^{2} - x^{2}}{a^{2}}$

$\Rightarrow y^{2} = \frac{b^{2}}{a^{2}} (a^{2} - x^{2})$

$\Rightarrow y = \pm \sqrt{\frac{b^{2}}{a^{2}} (a^{2} - x^{2})}$

$\Rightarrow y = \pm \frac{b}{a} \sqrt{(a^{2} - x^{2})}$

Since $O A B$ is in first quadrant, value of $y$ is positive.

$∴ y = \frac{b}{a} \sqrt{a^{2} - x^{2}}$

Area of ellipse = $4 \times \int_{0}^{a} y d x$

$= 4 \int_{0}^{a} \frac{b}{a} \sqrt{a^{2} - x^{2}} d x$

$= \frac{4 b}{a} \int_{0}^{a} \sqrt{a^{2} - x^{2}} d x$

It is of form $\sqrt{a^{2} - x^{2}} d x = \frac{1}{2} x \sqrt{a^{2} - x^{2}} + \frac{a^{2}}{2} \sin^{- 1} \frac{x}{a} + c$

$= \frac{4 b}{a} {[\frac{x}{2} \sqrt{a^{2} - x^{2}} + \frac{a^{2}}{2} \sin^{- 1} \frac{x}{a}]}_{0}^{a}$

$= \frac{4 b}{a} [(\frac{a}{2} \sqrt{a^{2} - a^{2}} + \frac{a^{2}}{2} \sin^{- 1} \frac{a}{a}) - (\frac{0}{2} \sqrt{a^{2} - 0} + \frac{a^{2}}{2} \sin^{- 1} (0))]$

$= \frac{4 b}{a} [0 + \frac{a^{2}}{2} \sin^{- 1} (1) - 0 - 0]$

$= \frac{4 b}{a} \times \frac{a^{2}}{2} \times \frac{π}{2}$

$= π a b$

$∴$ Required Area of Ellipse $= π a b$ sq.units

Odisha Board Solutions for Chapter: Area Under Plane Curves (Application of Definite Integrals), Exercise 1: EXERCISES-10

Odisha Board Mathematics Solutions for Exercise - Odisha Board Solutions for Chapter: Area Under Plane Curves (Application of Definite Integrals), Exercise 1: EXERCISES-10

Questions from Odisha Board Solutions for Chapter: Area Under Plane Curves (Application of Definite Integrals), Exercise 1: EXERCISES-10 with Hints & Solutions

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