Find the area of the region bounded by

Question:

Find the area of the region bounded by $y^{2}=9 x, x=2, x=4$ and the $x$-axis in the first quadrant.

Solution:

The area of the region bounded by the curve, $y^{2}=9 x, x=2$, and $x=4$, and the $x$-axis is the area ABCD.

Area of $\mathrm{ABCD}=\int_{2}^{1} y d x$

$=\int_{2}^{4} 3 \sqrt{x} d x$

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

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

$=2\left[(4)^{\frac{3}{2}}-(2)^{\frac{3}{2}}\right]$

$=2[8-2 \sqrt{2}]$

$=(16-4 \sqrt{2})$ units

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