Q 17. - Calculus

Question

Each of the sums in Exercises 17–20 approximates the area between the graph of some function f and the x-axis from x = a to x = b. Do some “reverse engineering” to determine the type of approximation (left sum, midpoint sum, etc.) and identify f(x), a, b, n, $∆x$, and $x_k$. Then sketch the approximation de-

scribed.

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

Step-by-Step Solution

Answer

$\begin{array}{rcl}f\left(x\right)& =& x^2\\ N& =& 4\\ ∆x& =& \frac{1}{2}\\ x_k& =& 1+\frac{k}{2}\\ a& =& 1\\ b& =& 3\end{array}$

Step 1. Given information

Function is $\sum _{k=1}^4{\left(1+\frac{k}{2}\right)}^2\left(\frac{1}{2}\right)$

Step 2. Explanation

$\sum _{k=1}^4{\left(1+\frac{k}{2}\right)}^2\left(\frac{1}{2}\right)={\left(1+\frac{1}{2}\right)}^2\left(\frac{1}{2}\right)+{\left(1+\frac{2}{2}\right)}^2\left(\frac{1}{2}\right)+{\left(1+\frac{3}{2}\right)}^2\left(\frac{1}{2}\right)+{\left(1+\frac{4}{2}\right)}^2\left(\frac{1}{2}\right)$

width="87" style="max-width: none; vertical-align: -69px;" $\begin{array}{rcl}f\left(x\right)& =& x^2\\ N& =& 4\\ ∆x& =& \frac{1}{2}\\ x_k& =& 1+\frac{k}{2}\end{array}$

Therefore,

$\begin{array}{rcl}a& =& 1\\ b& =& 3\end{array}$