I just have a quick question on quadratic equations, if we are given a whole bunch of data when plotted, it gives us some quadratic relationship. But I tried to use the old fashioned way to find this approximate equation by finding $c$ at $x =0,$ and: $$y_1=ax_1^2+bx_1+c$$ $$y_2=ax_2^2+bx_2+c$$ I come to find that the area of the curve in the excel from say $x_1$ to $x_{10}$ is significantly different to what I found using the curve that I have found using simultaneous equations integrating from $x_1$ to $x_{10}.$
So my question is how do we find a good approximate for a quadratic equation when we are given some set of data which exhibits a quadratic relationship?
You can try to do an OLS estimation. There's a function in Excel to do that, is called LINEAR.ESTIMATE or something similar. What this function does is finding the values for $a, b$ and $c$ that best fit your data. "Best fit" means, in this context, the values that make the $\sum_{i=1}^n (y_i - a x_{i}^2 - b x_i - c)^2$ as small as possible.