Integral partition for $f(x)=x^2$ on $[1,3]$

Question

I'm struggling with this question:

• If we have $f(x) = x^2 $ and $P_n $ which partitions $[1,3]$ into $n$ sub-intervals, each equal in length, how can I write the formulas for $L(f,P_n)$ and $U(f,P_n)$ in terms of $n$?

• Do they converge to the same limit as $n$ goes to infinity?

• How large does $n$ need to be so that $U(f,P_n)$ is within $0.01$ of $\int_1^3 \! f(x) \, \mathrm{d}x$ ?

Answer 1

The partition $P_n$ consists of the intervals $\left\{ \left[1+\frac{k-1}{n/2},1+\frac{k}{n/2} \right] \right\}_{k=1}^n$

Since $f(x)$ is monotonically increasing we have $$L(f,P_n)=\sum_{k=1}^nm_k \frac{2}{n}=\sum_{k=1}^n \left(1+\frac{k-1}{n/2} \right)^2 \frac{2}{n}, \\U(f,P_n)=\sum_{k=1}^n M_k \frac{2}{n}=\sum_{k=1}^n \left(1+\frac{k}{n/2} \right)^2 \frac{2}{n}.$$

This should help you a little.