$(a)$ Use the definition $to$ find $an$ expression for the area under the curve $y=x^3$ from $0to1as$ a $limit.$

$(b)$ The following formula for the sum $of$ the cubes $of$ the first $n$ integers $is$ proved $in$ Appendix $E$ :

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