The set $\{1,1.7,2,3.5,5\}$ determines a partition, $P$.
Let $z_1=1$, $z_2=2$, $z_3=3$, $z_4=4$, and $f(x)=\sqrt{4-\left(x-3\right)^2}$.
There’s three things I’m told to find:
$\lVert P\rVert$
The Riemann sum $R_P$ of $f$
And finally I need to find this mess: $$\lim\limits_{\lVert P\rVert \to 0} \sum_{i=1}^n \left(\sqrt{4-\left(z_i-3\right)^2}\right) \Delta x_i$$
I have the formula $$\int_a^b f(x) \,dx=\lim\limits_{\lVert P\rVert \to 0} \sum_{i=1}^n f\left(z_i\right) \Delta x_i$$ written down in my notes, but I’m not sure what to do with it (I’m not very strong with sigma notation).
I know that $\lVert P\rVert$ represents the largest subinterval in the set (the norm of $P$), and that this is $1.5$ (between the intervals $\left[2,3.5\right]$ and $\left[3.5,5\right]$). I’ve also calculated the values for each $z$ given to me:
$$f(z_1)=0$$ $$f(z_2)=\sqrt{3}$$ $$f(z_3)=2$$ $$f(z_4)=\sqrt{3}$$
I don’t know what to do with all this information. Any help with figuring this out is greatly appreciated.