Finding points along an inverted parabola curve

Question

I am doing a project on solar photovoltaic panels output. The total output on a sunny day is 244.6875 kWh (over 9 sun hours). Is there a way I can plot a typical inverted parabola curve and find the output between 2 specific points (maybe from point 3 to 4)?

Answer 1

I guess you are saying that output is $0$ at the beginning and end of the $9$-hour period, that it approximated by a parabola, and that the integral over the $9$-hour period is $244.6875.$ If we take the midpoint of the period to be $0$ then the output is of the form $$f(x)=C(x-4.5)(x+4.5)=C(x^2-20.25)$$ for some constant $C$.

We have $$\int_{-4.5}^{4.5}{C(x^2-20.25)dx}=244.6875\\ C\left(\frac{x^3}{3}-20.25x\right)\bigg\rvert_{-4.5}^{4.5}=244.6875\\ C\left(\frac{2(4.5)^3}{3}-9\cdot20.25\right)=244.6875\\ -121.5C=244.6875\implies C=-2.0138888\dots\\ $$ To find the output between two specific points, you just integrate, as above. The output from $a$ to $b$ is $$ -2.013889\int_a^b{(x^2-20.25)dx}=-2.013889\left(\frac{x^3}{3}-20.25x\right)\bigg\rvert_a^b $$ In computing $a$ and $b$ remember that the start of the period is at $-4.5,$ so if by "point $3$ to $4$" you mean the points $3$ and $4$ hours after the start of the $9$-hour period, then $a=-1.5, b=-.5.$