Question

find the area of the region bounded by the curves x=y³ and y=x²

Answer 1

We can rewrite the equation of the first curve $x=y^3$ as $\sqrt[3]{x}=y$ or $y=\sqrt[3]{x}.$

Let`s construct the graphical representation of the area:

The area of the region bounded by the curves equals a definite integral between the points of intersection of these curves: (0, 0) and (1, 1).

$A=\int_0^1 \sqrt[3]{x}dx - \int_0^1 x^2dx= \frac{3x\sqrt[3]{x}}{4}|_0^1 - \frac{x^3}{3}|_0^1=$

$=(\frac{3}{4}-0) - (\frac{1}{3}-0)=\frac{3}{4}-\frac{1}{3}=\frac{9}{12}-\frac{4}{12}=\frac{5}{12}.$

Answer: $\frac{5}{12}.$