5.5
Can somebody verify this solution for me? Thanks!
Find the area bounded by the curves $x^3$ and $x^4$ on $(0,1)$.
First we have to solve $x^3=x^4$ to find the bounds of integration.
$x^3=x^4$
$ \rightarrow 0 = x^4-x^3$
$\rightarrow 0 = x^3(x-1)$
and so $x=0$ and $x=1$ are solutions.
Furthermore, on the interval $(0,1)$ we have that $x^4<x^3$. The area between the graphs is thus:
$\int_0^1 x^3-x^4 dx$
$= \frac{x^4}{4}-\frac{x^5}{5}|_0^1$
$= \frac{1}{4} - \frac{1}{5} - 0 - 0$
$= \frac{1}{20}$