Integrate f(x)=2x. Lower limit is 1 and upper limit is 2.

Using the shortcuts, it's easy for me to find F(2) - F(1) = 3.

However, I was asked to use the limit of Reinmann sums in order to solve the problem...

$\displaystyle \int f(x) dx = \lim_{k\to\infty}\sum_{j=1}^{k}(f(x_j) \Delta x)$

$\displaystyle f(x) = 2x$

$\displaystyle \Delta x = \frac{2 - 1}{k} = \frac{1}{k}$

$\displaystyle x_j = \frac{1}{k} j = \frac{j}{k}$

$\displaystyle \sum_{j=1}^{k} 2 \frac{j}{k} (\frac{1}{k})$

$\displaystyle \frac{2}{k^2}\sum_{j=1}^{k}j$