Step $3$: What we accomplished with the last step was to turn complex formula, one that would be hard to apply a limit to$,$ into a formula which is relatively easy to find the limit of$.$ Find the exact area by taking the limit of this formula as the number of rectangles goes to infinity. That is$,$ find:

$\underset{n}{lim}(-\frac{128n^3+192n^2+64n}{6n^3}+\frac{48n^2+48n}{2n^2}-4)$

a$)$ Describe in words the role the limit plays in changing the approximate area into the exact area?

b$)$ To finish, apply the Fundamental Theorem of Calculus to find the area using an antiderivative. Be sure the answer you get matches with what you found above.

${\displaystyle \underset{a}{\overset{b}{}}}f(x)dx=F(b)-F(a)$

where $F(x)$ is an antiderivative of $f(x).$