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Question
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.
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
\[= \int_a^{2a} x_{\text{ curve }} dy\]
\[ = \int_a^{2a} \left( a y^2 \right)^\frac{1}{3} dy\]
\[ = a^\frac{1}{3} \int_a^{2a} y^\frac{2}{3} dy\]
\[ = \left.a^\frac{1}{3} \times \frac{y^\frac{5}{3}}{\frac{5}{3}}\right|_a^{2a} \]
\[ = \frac{3}{5} a^\frac{1}{3} \left[ \left( 2a \right)^\frac{5}{3} - a^\frac{5}{3} \right]\]
\[ = \frac{3}{5}\left( 2^\frac{5}{3} a^2 - a^2 \right)\]
\[ = \frac{3}{5}\left( 2^\frac{5}{3} - 1 \right) a^2\text{ square units }\]
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