Hello Belinda Obeng,
First, let's determine the points of intersection for the two curves. Substituting for $y$ as given in the first equation into the second, we obtain:
$$\left(x^2 \right)^2=8x$$
$$x^4-8x=0$$
$$x\left(x^3-2^3 \right)=0$$
$$x(x-2)\left(x^2+2x+2^2 \right)=0$$
The quadratic factor has complex roots, hence we find:
$$x=0,\,2$$
Thus, the two points of intersection are:
$$(0,0),\,(2,4)$$
To use the washer method, we find the volume of an arbitrary washer is:
$$dV=\pi\left(R^2-r^2 \right)\,dy$$
where:
$$R^2=y$$
$$r^2=\frac{y^4}{64}$$
Hence:
$$dV=\pi\left(y-\frac{y^4}{64} \right)\,dy$$
Adding the volume elements by integrating, we find:
$$V=\pi\int_0^4 y-\frac{y^4}{64}\,dy$$
Applying the FTOC, we obtain:
$$V=\pi\left[\frac{y^2}{2}-\frac{y^5}{320} \right]_0^4=\frac{24\pi}{5}$$
Using the shell method, we find the volume of an arbitrary shell is:
$$dV=2\pi rh\,dx$$
where:
$$r=x$$
$$h=\sqrt{8x}-x^2$$
Hence:
$$dV=2\pi x\left(\sqrt{8x}-x^2 \right)\,dx=2\pi\left(2\sqrt{2}x^{\frac{3}{2}}-x^3 \right)\,dx$$
Summing the volume elements by integrating, we find:
$$V=2\pi\int_0^2 2\sqrt{2}x^{\frac{3}{2}}-x^3\,dx$$
Applying the FTOC, we find:
$$V=2\pi\left[\frac{4\sqrt{2}}{5}x^{\frac{5}{2}}-\frac{x^4}{4} \right]_0^2=2\pi\left(\frac{32}{5}-4 \right)=\frac{24\pi}{5}$$
