$($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$.$ of approximating rectangles.

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