Chapter 11, Problem 20STP

To determine

To find: The formula for exact area under the graph.

Answer to Problem 20STP

(A) $2\underset{n\to \infty }{\lim }{\displaystyle \sum _{i=1}^{n}3}{\left(\frac{ki}{n}\right)}^2\left(\frac{k}{n}\right)$ is the formula for exact area under the graph of $f\left(x\right)=3x^2$ on the interval $\left[-k,k\right]$ .

Explanation of Solution

Given information: $f\left(x\right)=3x^2$ on the interval $\left[-k,k\right]$ .

(A) $2\underset{n\to \infty }{\lim }{\displaystyle \sum _{i=1}^{n}3}{\left(\frac{ki}{n}\right)}^2\left(\frac{k}{n}\right)$

(B) $2\underset{n\to \infty }{\lim }{\displaystyle \sum _{i=1}^n{\left(-k+\frac{2ki}{n}\right)}^2\left(\frac{k}{n}\right)}$

(C) $\underset{n\to \infty }{\lim }{\displaystyle \sum _{i=1}^{n}3}{\left(\frac{ki}{n}\right)}^2\left(\frac{k}{n}\right)$

(D) $\underset{n\to \infty }{\lim }{\displaystyle \sum _{i=1}^{n}3}{\left(-k+\frac{2ki}{n}\right)}^2\left(\frac{k}{n}\right)$

Formula used: Let $f$ be a function that is continuous and nonnegative on the interval $\left[a,b\right]$ . The area $A$ of the region bounded by the graph of $f$ , the x -axis, and the vertical lines $x=a$ and $x=b$ is given by

  $A=\underset{n\to \infty }{\lim }{\displaystyle \sum _{i=1}^{n}f\left(a+\frac{\left(b-a\right)i}{n}\right)\left(\frac{b-a}{n}\right)}$

Calculation: For $\left[0,k\right]$

Dimensions of rectangle:

Width: $\frac{b-a}{n}=\frac{k-0}{n}=\frac{k}{n}$

Height: $f\left(a+\frac{\left(b-a\right)i}{n}\right)=f\left(0+\frac{\left(k-0\right)i}{n}\right)=f\left(\frac{ki}{n}\right)=3{\left(\frac{ki}{n}\right)}^2$

Area of region under the graph of $f\left(x\right)=3x^2$ on the interval $\left[-k,k\right]$ .

  $A=2\underset{n\to \infty }{\lim }{\displaystyle \sum _{i=1}^{n}f\left(a+\frac{\left(b-a\right)i}{n}\right)\left(\frac{b-a}{n}\right)}$

  $=2\underset{n\to \infty }{\lim }{\displaystyle \sum _{i=1}^{n}3{\left(\frac{ki}{n}\right)}^2\left(\frac{k}{n}\right)}$

Thus, (A) $2\underset{n\to \infty }{\lim }{\displaystyle \sum _{i=1}^{n}3}{\left(\frac{ki}{n}\right)}^2\left(\frac{k}{n}\right)$ is the formula for exact area under the graph of $f\left(x\right)=3x^2$ on the interval $\left[-k,k\right]$ .