Chapter 13.1, Problem 29E

(a)

To determine

To calculate: A formula for the sum of the areas of the n rectangles under the curve of the function $y=x^2$ from $x=0$ to $x=1$ where the function is evaluated at right-hand end points of each subinterval.

(b)

To determine

To calculate: The value of $S\left(10\right)$ using the formula $S=\frac{\left(n+1\right)\left(2n+1\right)}{6n^2}$ for the sum of the areas of the n rectangles.

(c)

To determine

To calculate: The value of $S\left(100\right)$ using the formula $S=\frac{\left(n+1\right)\left(2n+1\right)}{6n^2}$ for the sum of the areas of the n rectangles.

(d)

To determine

To calculate: The value of $S\left(1000\right)$ using the formula $S=\frac{\left(n+1\right)\left(2n+1\right)}{6n^2}$ for the sum of the areas of the n rectangles.

(e)

To determine

To calculate: The value of $\underset{n\to \infty }{\lim }S$ where $S=\frac{\left(n+1\right)\left(2n+1\right)}{6n^2}$ is the formula for the sum of the areas of the n rectangles under the curve of the function $y=x^2$ from $x=0$ to $x=1$.