Question about Riemann sum and right endpoints

Question

I’ve gotten every question except the last two right. I’m just wondering if someone could explain to me how I do them.

Answer 1

You have all the formulas, you just need to apply them.

• $\Delta x = \frac{11 - 3}{n} = \frac8n$
• $x_k = 3 + k \Delta x = 3 + \frac{8k}n$
• $f(x_k) = (3 + \frac{8k}n)^2 = 9 + \frac{48 k}{n} + \frac{64 k^2}{n^2}$
• $f(x_k)\Delta x = \left( 9 + \frac{48 k}{n} + \frac{64 k^2}{n^2} \right) \frac8n $
• $\displaystyle\sum_{k=1}^n f(x_k)\Delta x = \sum_{k = 1}^n \left( 9 + \frac{48 k}{n} + \frac{64 k^2}{n^2} \right) \frac8n = \frac8n \left( 9 \sum_{k = 1}^n 1 + \frac{48}{n} \sum_{k = 1}^nk +\frac{64}{n^2} \sum_{k = 1}^nk^2 \right) $ (Use the formulas in your textbook for these special sums.)

A formula like $\sum_{k=1}^n f(x_k)\Delta x$ might seem scary at first but you just have to break it down into small pieces: $x_k$, $\Delta x$, $f(x_k)$ and then work your way up to the whole thing. That's the purpose of the first several parts of the problem: breaking down all the steps.