Name: Area Enclosed by a Curve and a Line: Illustration
Uploaded: 2019-07-19
Duration: 3 min 35 s
Description: How would you find the area enclosed between a parabola and a straight line? Watch the video to see the method as an application of definite integrals.

Question

Find the area of the circle

4 x^{2} + 4 y^{2} = 9

, which is interior to the parabola

y^{2} = 4 x

.

Accepted Answer

Given equations are $4 x^{2} + 4 y^{2} = 9 \Rightarrow x^{2} + y^{2} = {(\frac{3}{2})}^{2}$ and $y^{2} = 4 x$

To find points of intersection of given curves, substitute $y^{2} = 4 x$ in $x^{2} + y^{2} = {(\frac{3}{2})}^{2}$

$\Rightarrow x^{2} + 4 x = \frac{9}{4}$

$\Rightarrow 4 x^{2} + 16 x = 9$

$\Rightarrow 4 x^{2} + 16 x - 9 = 0$

$\Rightarrow 4 x^{2} + 18 x - 2 x - 9 = 0$

$\Rightarrow (2 x + 9) (2 x - 1) = 0$

$\Rightarrow x = \frac{1}{2}, \frac{- 9}{2}$ (rejected as parabola is symmetric about positive $x$ -axis)

Required sketch

Circle is symmetrical about both axes and parabola is symmetrical about $x$ -axis.

Therefore, the required area $= 2 (\int_{0}^{\frac{1}{2}} y_{1} d x + \int_{\frac{1}{2}}^{\frac{3}{2}} y_{2} d x)$

${y_{1} = \sqrt{4 x} d x, y_{2} = \sqrt{{(\frac{3}{2})}^{2} - x^{2}} d x}$

$= 2 [\int_{0}^{\frac{1}{2}} \sqrt{4 x} d x + \int_{\frac{1}{2}}^{\frac{3}{2}} \sqrt{{(\frac{3}{2})}^{2} - x^{2}} d x]$

$= 2 [2 {(\frac{x^{\frac{3}{2}}}{\frac{3}{2}})}_{0}^{\frac{1}{2}} + {(\frac{x}{2} \sqrt{{(\frac{3}{2})}^{2} - x^{2}} + \frac{{(\frac{3}{2})}^{2}}{2} \sin^{- 1} (\frac{x}{\frac{3}{2}}))}_{\frac{1}{2}}^{\frac{3}{2}}]$

$= \frac{8}{3} {(x^{\frac{3}{2}})}_{0}^{\frac{1}{2}} + {(x \sqrt{\frac{9}{4} - x^{2}} + \frac{9}{4} \sin^{- 1} \frac{2 x}{3})}_{\frac{1}{2}}^{\frac{3}{2}}$

$= \frac{8}{3} (\frac{1}{2 \sqrt{2}} - 0) + [(0 + \frac{9}{4} \cdot \frac{π}{2}) - (\frac{1}{2}) \sqrt{2} - \frac{9}{4} \sin^{- 1} \frac{1}{3}]$

$= \frac{\sqrt{2}}{6} + \frac{9}{4} (\frac{π}{2} - \sin^{- 1} \frac{1}{3}) sq . units$

K. C. Sinha Solutions for Chapter: Areas Enclosed by Curves (Applications of Integrals), Exercise 1: EXERCISE

K. C. Sinha Mathematics Solutions for Exercise - K. C. Sinha Solutions for Chapter: Areas Enclosed by Curves (Applications of Integrals), Exercise 1: EXERCISE

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