Answer
The area of the region is $48/5$.
Work Step by Step
$y=x^4$ and $y=8x$
1) Draft the graph
The graph is enclosed below. From the graph, we choose integration with respect to $x$ to employ here for calculation of the region area.
2) Find the limits of integration:
We can find the limits of integration by finding points of intersection between the curve and the line.
$$x^4-8x=0$$ $$x(x^3-8)=0$$ $$x(x-2)(x^2+2x+4)=0$$ $$x=0\hspace{1cm}\text{or}\hspace{1cm}x=2$$
So, the upper limit is $2$ and the lower one is $0$.
3) Find the area:
Looking at the draft, we see that the region from $x=0$ to $x=2$ is bounded above by $y=8x$ and below by $y=x^4$. So, according to definition, the area of the region is
$$A=\int^2_{0}[8x-x^4]dx$$ $$A=4x^2-\frac{x^5}{5}\Big]^2_{0}$$ $$A=4\times2^2-\frac{2^5}{5}$$ $$A=16-\frac{32}{5}=\frac{48}{5}$$
