Q 19. - Calculus

Question

The following sum 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 described.

$\sum _{k=1}^{100}\frac{\sin \left(0.05\left(k-1\right)+\sin \left(0.05k\right)\right)}{2}\left(0.05\right)$

Step-by-Step Solution

Answer

$\begin{array}{rcl}f\left(x\right)& =& \sin x\\ N& =& 100\\ ∆x& =& 0.05\\ x_k& =& 0+0.05\\ a& =& 0\\ b& =& 5\end{array}$

Step 1. Given information

Sum is $\sum _{k=1}^{100}\frac{\sin \left(0.05\left(k-1\right)+\sin \left(0.05k\right)\right)}{2}\left(0.05\right)$

Step 2. Explanation

Consider the sum $\sum _{k=1}^{100}\frac{\sin \left(0.05\left(k-1\right)+\sin \left(0.05k\right)\right)}{2}\left(0.05\right)$

$\begin{array}{rcl}f\left(x\right)& =& \sin x\\ N& =& 100\\ ∆x& =& 0.05\\ x_{k-1}& =& 0+0.05(k-1)\\ x_k& =& 0+0.05\\ a& =& 0\\ b& =& 0+0.05\left(100\right)\\ & =& 5\end{array}$