Find the area of the region bounded by the curve $a y^{2} = x^{3}$ , the y-axis and the lines y = a and y = 2a.

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Solution

The equation of the given curve is $a y^{2} = x^{3}$ .

The given curve passes through the origin. This curve is symmetrical about the x-axis.

The graph of the given curve is shown below.

The lines y = a and y = 2a are parallel to the x-axis and intersects the y-axis at (0, a) and (0, 2a), respectively.

∴ Required area = Area of the shaded region

$= \int_{a}^{2 a} x_{curve} d y = \int_{a}^{2 a} {(a y^{2})}^{\frac{1}{3}} d y = a^{\frac{1}{3}} \int_{a}^{2 a} y^{\frac{2}{3}} d y = a^{\frac{1}{3}} \times {\frac{y^{\frac{5}{3}}}{\frac{5}{3}}}_{a}^{2 a} = \frac{3}{5} a^{\frac{1}{3}} [{(2 a)}^{\frac{5}{3}} - a^{\frac{5}{3}}] = \frac{3}{5} (2^{\frac{5}{3}} a^{2} - a^{2}) = \frac{3}{5} (2^{\frac{5}{3}} - 1) a^{2} square units$

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