Find the area of the region bounded by the curve
Question:

Find the area of the region bounded by the curve $y^{2}=4 x$ and the line $x=3$

Solution:

The region bounded by the parabola, $y^{2}=4 x$, and the line, $x=3$, is the area $\mathrm{OACO}$.

The area OACO is symmetrical about x-axis.

∴ Area of OACO = 2 (Area of OAB)

Area $\mathrm{OACO}=2\left[\int_{0}^{3} y d x\right]$

$=2 \int_{0}^{3} 2 \sqrt{x} d x$

$=4\left[\frac{x^{\frac{3}{2}}}{\frac{3}{2}}\right]_{0}^{3}$

$=\frac{8}{3}\left[(3)^{\frac{3}{2}}\right]$

$=8 \sqrt{3}$

Therefore, the required area is $8 \sqrt{3}$ units.

 

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