Answer
$\displaystyle \int_{0}^{4}6x(4-x)^{2}dx$
Work Step by Step
Definition of Definite lntegral
If $f$ is defined on the closed interval $[a, b]$
and the limit of Riemann sums over partitions $\Delta$
$\displaystyle \lim_{\Vert\Delta||\rightarrow 0}\sum_{i=1}^{n}f(c_{i})\Delta x_{i}$
exists (as described above),
then $f$ is said to be integrable on $[a, b]$ and the limit is denoted by
$\displaystyle \lim_{\Vert\Delta\Vert\rightarrow 0}\sum_{i=1}^{n}f(c_{i})\Delta x_{i}=\int_{a}^{b}f(x)dx$.
The limit is called the definite integral of $f$ from $a$ to $b$.
The number $a$ is the lower limit of integration, and the number $b$ is the upper limit of integration.
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Comparing with the definition:
The closed interval: $[a, b]$ = $[0,4]$ .
Substitute $c_{i}$ with $x$: $f(x)=6x(4-x)^{2}$
If the limit is defined, then we write,
$\displaystyle \lim_{\Vert\Delta]\rightarrow 0}\sum_{i=1}^{n}(6c_{i}(4-c_{i})^{2})\Delta x_{i}=\int_{0}^{4}6x(4-x)^{2}dx$
