Q. 7TF - Calculus

Question

Limits of Riemann sums: In the reading we saw that the limits of 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 $A\left(n\right)$:

What does your graph tell you about the right-sum approximations of the area under the graph of $f$ 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 values 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 areas of the $n$ rectangles.

$A\left(n\right)=\sum _{k-1}^n(1+\frac{4k^2}{n^2})\left(\frac{2}{n}\right)$

This is the right sum approximation to find the area under the curve.

As $n\to \infty$
$\lim A\left(n\right)=4.67$
$n\to \infty$