How does the riemann sum definition work?

Question

I've been looking up the limit definitions of Riemann sums and integrals for the past half hour but could find nothing that helped me on this question. what does the K do? and how to find the upper and lower integrations?

Thanks in advance!

Answer 1

Here, $k$ is serving the same purpose as the $i$ was in your other thread. The stuff involving $k$ in the sum is basically your $f(x_k^*)$, and the number that it's multiplied by (here, $\frac{1}{2}$ for each term) is the length of the interval. Since the problem states that the sum is a right approximation, then we have $x_k^* = x_{k}$ (we know that $x_k^*$ is in $[x_{k-1}, x_{k}]$; this just tells us that $x_k^*$ is all the way to the right of the interval).

To find the upper and lower limits, note that for $k=0,1,2,3,4,5,6$ we have $\frac{k+4}{2} = 2,2.5,3,2.5,4,4.5,5$, and so the integral is taken on $[2,5]$. The reason I calculate $\frac{k+4}{2}$ is because that's what was substituted for $x$ in $\sqrt[3]{x}+4x$, which you could probably guess was the correct integrand.