Q. 40 - 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.

$f\left(x\right)=x^2,\left[a,b\right]=\left[0,3\right],left sum,n=3$

Step-by-Step Solution

Answer

The right sum is $\sum _{k=1}^3{k}^2$

Step 1. Given Information

The given function is $f\left(x\right)=x^2,\left[a,b\right]=\left[0,3\right],left sum,n=3$

Step 2. Finding ∆ x and x k

$∆x=\frac{b-a}{n}=\frac{3-0}{3}=1$ and $x_k=a+k∆x=0+k=k$

Step 3. Finding Reimann Sum

We have $f\left(x\right)=x^2$then $f\left(x_{k-1}\right)={\left(x_{k-1}\right)}^2$

The Reimann Sum is

 $$\begin{gathered} =\sum _{k=1}^{n}f\left(x_{k-1}\right)∆x \\ =\sum _{k=1}^{3}f\left(k\right)1 \\ =\sum _{k=1}^3{k}^2 \end{gathered}$$