Q. 43 - Calculus

Question

In Exercises 39–44, write out the sigma notation for the Riemann sum described in such a way that the only letter which appears in the general term of the sum is k. Don’t calculate the value of the sum; just write it down in sigma notation.

$f\left(x\right)=\sin x,\left[a,b\right]=\left[0,\mathrm{\pi }\right],trapezoid sum,n=4$

Step-by-Step Solution

Answer

The trapezoid sum is$\frac{\mathrm{\pi }}{4} \sum _{k=1}^4\frac{\sin \left({\displaystyle \frac{\mathrm{\pi }\left(\mathrm{k}+1\right)}{4}}\right)+\sin \left(\frac{\mathrm{\pi k}}{4}\right)}{2}$

Step 1. Given Information

We have $f\left(x\right)=\sin x,\left[a,b\right]=\left[0,\mathrm{\pi }\right],trapezoid sum,n=4$

Step 2. Finding ∆ x , x k and x k + 1

$∆x=\frac{b-a}{n}=\frac{\mathrm{\pi }-0}{n}=\frac{\mathrm{\pi }}{4}$

$$\begin{gathered} x_k=a+k∆x=\frac{\mathrm{\pi k}}{4} \\ {x}_{k+1}=a+\left(k+1\right)∆x=\frac{\pi (k+1)}{4} \end{gathered}$$

Step 3. Finding Riemann Sums

We have $f\left(x\right)=\sin x$ then $$\begin{gathered} f\left(x_{k+1}\right)=f\left(\frac{\mathrm{\pi }\left(\mathrm{k}+1\right)}{4}\right)=\sin \left(\frac{\mathrm{\pi }\left(\mathrm{k}+1\right)}{4}\right) \\ f\left(x_k\right)=f\left(\frac{\mathrm{\pi k}}{4}\right)=\sin \left(\frac{\mathrm{\pi k}}{4}\right) \end{gathered}$$

and $\frac{f\left(x_{k+1}\right)+f\left(x_k\right)}{2}=\frac{\sin \left({\displaystyle \frac{\mathrm{\pi }\left(\mathrm{k}+1\right)}{4}}\right)+\sin \left(\frac{\mathrm{\pi k}}{4}\right)}{2}$

Now, Riemann Sum

$$\begin{gathered} =\sum _{k=1}^n\frac{f\left(x_{k+1}\right)+f\left(x_k\right)}{2}∆x \\ =\frac{\mathrm{\pi }}{4} \sum _{k=1}^4\frac{\sin \left({\displaystyle \frac{\mathrm{\pi }\left(\mathrm{k}+1\right)}{4}}\right)+\sin \left(\frac{\mathrm{\pi k}}{4}\right)}{2} \end{gathered}$$