When approximating the area beneath the curve $f(x)=x^2+1$ on the interval $[0,3]$ using a left Riemann sum with 4 rectangles, I calculated that the height of each rectangle, in order from left to right, would be
$f(0)$, $f(\frac{3}{4})$, $f(\frac{3}{2})$, $f(\frac{9}{4})$
and therefore, the area would be
$A=\frac{3}{4}\cdot f(0)+\frac{3}{4}\cdot f(\frac{3}{4})+\frac{3}{4}\cdot f(\frac{3}{2})+\frac{3}{4}\cdot f(\frac{9}{4})$ (I'll call this equation 1)
Now I know the next step is
$A=\frac{3}{4}\cdot1+\frac{3}{4}\cdot \frac{25}{16}+\frac{3}{4}\cdot \frac{13}{4}+\frac{3}{4}\cdot\frac{97}{16}=\frac{285}{32}\approx8.906$ (I'll call this equation 2)
But it's this step that confuses me.
How does one know for example that $f(\frac{3}{4})$ from equation 1 goes to $\frac{25}{16}$ in equation 2 and $f(\frac{9}{4})$ from equation 1 goes to $\frac{97}{16}$ in equation 2? (The same goes for the other variables).
I apologise if this is written in a confusing way.