Q. 6TF - Calculus

Question

Limits of Riemann sums: In the reading we saw that the

area between the graph of $f\left(x\right)=x^2-2x+2$ and the

$x-$axis on $\left[1,3\right]$ could be approximated with the right sum

$\sum _{k=1}^n\left(1+\frac{4k^2}{n^2}\right)\left(\frac{2}{n}\right)$. Let $A\left(n\right)$ be equal to this n-rectangle right-sum approximation. The following table describes various values of A(n):

Use the table to make a graph of $A\left(n\right)$, and discuss what happens to this graph as n approaches infinity.

Step-by-Step Solution

Answer

As the number of rectangles goes to infinity the area under the function becomes a constant.

Step 1. Given Information

Table with various vales of $A\left(n\right)$:

Step 2. The Graph

The graph of $A\left(n\right)$:

The area under the curve $f\left(x\right)=x^2-2x+2$ on the interval $\left[1,3\right]$ is given by the sum of area of the $n$rectangles.
$A\left(n\right)=\sum _{k=1}^n\left(1+\frac{4k^2}{n^2}\right)\left(\frac{2}{n}\right)$
As$n\to \infty$

$\lim A\left(n\right)=4.67$

$n\to \infty$