CPM Homework Help : APCALC Problem 4-60

Change the following limit of a Riemann sum for the area under a curve into an integral expression. Then, evaluate the sum with your graphing calculator.
$\lim\limits _ { n \rightarrow \infty } \sum _ { i = n } ^ { 5 n } \frac { 1 } { n } ( ( \frac { i } { n } ) ^ { 3 } - 4 ) = \int _ { - } ^ { - } ( \underline { } ) _ { - }$

Hint:

This Riemann sum can be interpreted as computing the areas of infinitely many rectangles with:

$\text{base}=\lim \limits_{n\to\infty}\frac{1}{n}\text{ and height}=\Big(\frac{i}{n}\Big)^3-4$

Can you rewrite the given Riemann sum as an integral?

Step 1:

$f(x)dx=(x^3-4)dx$

Step 2:

Use the above statement and the bounds of the sum to determine the bounds of the integral.

Answer:

$\int_{1}^{5}\left ( x^{3}-4 \right)dx=?$