The area of the region bounded by the curve $y^2 = x^3$, the y-axis and the lines $y = 1$ and $y = 8$ is
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When calculating areas bounded by the y-axis and horizontal lines ($y=c$ to $y=d$), always rewrite the function as $x = f(y)$ and integrate with respect to $dy$. This avoids having to split the integral into multiple parts or dealing with more complex inverse functions if you tried integrating with respect to $dx$.