Chapter 13.1, Problem 27E

(a)

To determine

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

(b)

To determine

To calculate: The value of $S\left(10\right)$ using the formula $S=\frac{\left(n-1\right)}{n}$ 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 $S=\frac{\left(n-1\right)}{n}$ 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 $S=\frac{\left(n-1\right)}{n}$ 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 $S=\frac{\left(n-1\right)}{n}$ is the formula for the sum of the areas of the n rectangles under the curve of the function $y=2x$ from $x=0$ to $x=1$.