Q. 39 - 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.

$f\left(x\right)=\sqrt{x-1},\left[a,b\right]=\left[2,3\right],right sum,n=4$

Step-by-Step Solution

Answer

The right sum is given by $0.25\sum _{k=1}^4\sqrt{\left(1+0.25k\right)}$$0.25\sum _{k=1}^4\sqrt{\left(1+0.25k\right)}$

Step 1. Given Information

The given function is $f\left(x\right)=\sqrt{x-1},\left[a,b\right]=\left[2,3\right],right sum,n=4$

Step 2. Finding ∆ x and x k

$∆x=\frac{b-a}{n}=\frac{3-2}{4}=\frac{1}{4}=0.25$ and $x_k=a+k∆x=2+0.25k$

Step 3. Finding Reimann Sum

We have $f\left(x\right)=\sqrt{x-1}$then $f\left(x_k\right)=\sqrt{x_k-1}$

$$\begin{gathered} \sum _{k=1}^{n}f\left(x_k\right)∆x=\sum _{k=1}^{4}f\left(2+0.25k\right)0.25 \\ =0.25\sum _{k=1}^{4}f\left(2+0.25k\right)=0.25\sum _{k=1}^4\sqrt{\left(2+0.25k\right)-1} \\ =0.25\sum _{k=1}^4\sqrt{\left(1+0.25k\right)} \end{gathered}$$