I have been given this rather simple looking assignment, which is confusing me a lot.
Given is: $f:[0,5] \rightarrow \mathbb{R}$ where $f(x)=2x+3$.
The first thing I had to do, was to determine the area under the curve, for the given interval $[0,5]$, from the left and right side, for 30 subintervals. I solved this by using $$ S_J = \sum_{j = 1}^{J} f(\xi) \Delta x_j $$
What I now have to solve is: Find the exact values for the riemann-summ in the interval $[0,5]$ with $n$ subintervals, from the left and right side
My initial thought was to do this, to find the exact area:
$$ S_J = \int_{a}^{b} f(x) \, dx $$ But that how would that allow me to do it from the left and right side?
Any ideas? I am completely lost
If you divide the interval $[0,5]$ into $n$ equal subintervals, they have endpoints at $x_j = \frac{5j}{n}$, for each $j$ from $0$ to $n$.
So you have to evaluate the sums $$\begin{aligned} L_J &= \sum_{j=1}^J f(x_{j-1}) \frac{5}{n} \\ U_J &= \sum_{j=1}^J f(x_j) \frac{5}{n} \end{aligned}$$ The idea is to find a closed-form expression for the sum in terms of $J$ alone. If this kind of series is unfamiliar to you, you might consider reading about sums of arithmetic progressions (from Wikipedia) or researching the formula: $$ \sum_{j=1}^J j = \frac{J(J+1)}{2} $$