Question #345127

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

1
Expert's answer
2022-05-26T17:19:25-0400
y=x3,y=x2y=\sqrt[3]{x}, y=x^2

x3=x2\sqrt[3]{x} =x^2

x=x6x=x^6

x(x51)=0x(x^5-1)=0

x1=0,x2=1x_1=0, x_2=1

A=01(x3x2)dxA=\displaystyle\int_{0}^{1}(\sqrt[3]{x} -x^2)dx=[3x4/34x33]10=\big[\dfrac{3x^{4/3}}{4}-\dfrac{x^3}{3}\big]\begin{matrix} 1 \\ 0 \end{matrix}=3413=512(units2)=\dfrac{3}{4}-\dfrac{1}{3}=\dfrac{5}{12}({units}^2)


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