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  4. The area of the region (in sq. units), in the first quadrant bounded by the parabola y = 9x2 and the lines x = 0, y = l and y = 4, is :

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Q. The area of the region (in sq. units), in the first quadrant bounded by the parabola $y = 9x^2$ and the lines $x = 0, y = l$ and $y = 4$, is :

Application of Integrals

Solution:

Required area $= \int\limits^{4}_{y = 1}\sqrt{\frac{y}{9}} dy$
$ = \frac{1}{3} \int\limits^{4}_{y = 1}y^{1/2} dy = \frac{1}{3} \times \frac{2}{3} \left(y^{3/2}\right)|^{4}_{1}$
$= \frac{2}{9}\left[\left(4^{1/2}\right)^{3}-\left(1^{1/2}\right)^{3}\right] = \frac{2}{9}\left[8-1\right] = \frac{2}{9} \times 7 = \frac{14}{9}$ sq. units.