35-36 Use a graph to find approximate [tex]x[/tex] -coordinates of the points of intersection of the given curves. Then use your calculator to find (approximately) the volume of the solid obtained by rotating about the [tex]x[/tex] -axis the region bounded by these curves.
36. [tex]y=\sqrt[3]{2 x-x^{2}}, \quad y=x^{2} /\left(x^{2}+1\right)[/tex]

The approximate x-coordinates of the points of intersection of the given curves are 0 & 1.76, and the volume of the solid when rotated about the x-axis is 3.15 cubic units

Part A: The points of intersection

The equations are given as:

y = ∛2x - x²

y = x²/(x² + 1)

See attachment for the graph of the above curves (equation)

From the attached curve, we have the following points of intersection

(x,y) = (0,0) and (1.76,0.76)

Remove the y coordinates

x = 0 and 1.76

Hence, the approximate x-coordinates of the points of intersection of the given curves are 0 and 1.76

The volume of the solid when rotated about the x-axis

The volume of the solid is calculated using:

V = π(R² - r²)

Where R and r are the outer and the inner radii, respectively.

In this case;

R = ∛2x - x²

r = x²/(x² + 1)

Using a calculator, we have:

V = 3.15

Hence, the volume of the solid when rotated about the x-axis is 3.15 cubic units