$[0/4$ Points$]$

$($a$)$ Use the definition given below with right endpoints to express the area under the curve $y=x^3$ from $0$ to $1$ as a limit$.$

$A=\underset{n}{lim}{R}_n=\underset{n}{lim}[f(x_1)x+f(x_2)x+$cdots$+f(x_n)x]=\underset{n}{lim}{\displaystyle \underset{i=1}{\overset{n}{}}}f(x_i)x$

$\underset{n}{lim}{\displaystyle \underset{i=1}{\overset{n}{}}}()$

$($b$)$ Use the following formula for the sum of cubes of the first $n$ integers to evaluate the limit in part $($a$).$

$1^3+2^3+3^3+$cdots$+n^3=[\frac{n(n+1)}{2}{]}^2$

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