Answer to Question #345127 in Calculus for Jane

Question

Find the area of the region bounded by the curves X= Y³ and Y= X²

Answer 1

$$y=\sqrt[3]{x}, y=x^2$$

$$\sqrt[3]{x} =x^2$$

$$x=x^6$$

$$x(x^5-1)=0$$

$$x_1=0, x_2=1$$

$$A=\displaystyle\int_{0}^{1}(\sqrt[3]{x} -x^2)dx$$ $$=\big[\dfrac{3x^{4/3}}{4}-\dfrac{x^3}{3}\big]\begin{matrix} 1 \\ 0 \end{matrix}$$ $$=\dfrac{3}{4}-\dfrac{1}{3}=\dfrac{5}{12}({units}^2)$$