The following sum
$$\frac{7}{n}\sqrt{49-\left(\frac{7}{n}\right)^2}+\frac{7}{n}\sqrt{49-\left(\frac{14}{n}\right)^2}+ \cdots+\frac{7}{n}\sqrt{49-\left(\frac{7n}{n} \right)^2}$$
is a Riemann sum with $n$ subintervals of equal length for the definite integral
$$\int f(x)\,dx$$
where $b =$
and $f(x) =$
The integration should be $b \to 0$ but I couldn't figure it out. I solved for $b$ which I got $b=7$ and then solved for $f(x)$ which was $f(x)=\sqrt{49-(x-7)^2}$. I got $b$ correct and $f(x)$ wrong. I used this to help solve. Here is all my rough work:
Any help is appreciated
\begin{align}&\sqrt{(49-\left(\frac{7}{n}\right)^2}\left(\frac{7}{n}\right)+\sqrt{49-\left(\frac{14}{n}\right)^2}\left(\frac{7}{n}\right)+\ldots+\sqrt{49-\left(\frac{7n}{n}\right)^2}\left(\frac{7}{n}\right)\\&=\left( \frac7n\right)\sum_{i=1}^n \sqrt{49-\left( \frac{7i}{n}\right)^2} \end{align}
You have managed to identified that $\Delta x= \frac{7}{n}$
Hence we have $$\left( \frac7n\right)\sum_{i=1}^n \sqrt{49-\left( \frac{7i}{n}\right)^2} =\Delta x \sum_{i=1}^n \sqrt{49-(i\Delta x)^2}$$
Can you identify $f$ now?