I have the next function of temperature ($T$): $$f(T)=a+bT+cT^2+dT^3+eT^4$$ (where $a$,$b$,$c$,$d$,$e$ are known constants). I need to solve this equation: $$\int_{T_0}^{T_1}f(T)dT=1$$ The only unknown is $T_1$, which needs to be found ($T_0$ is known).
I need to do this numerically, using Matlab. However I don't have the Symbolic Math Toolbox, nor the Optimization Toolbox (I can't make use of fsolve or solve). Even if I had access to these two Toolboxes, I wouldn't know how to approach the problem.
Any ideas or help will be well received.
Let $$F(t)=aT+b \tfrac{T^2}{2}+c \tfrac{T^3}{3}+d \tfrac{T^4}{4}+e \tfrac{T^5}{5}.$$
be a primitive function of $f$, meaning that $$\int_{T_0}^{T_1}f(t)dt=F(T_1)-F(T_0).$$
Thus basicaly, you are looking for a solution in $T_1$ such that $F(T_1)=F(T_0)+1$ where $F(T_0)$ is know (i.e., is a constant).
Thus, the simplest thing would be
a) first to plot the graphical representation of $F$ and know where is situated the solution, more exactly "bracket it" between two extreme values.
b) then use a dichotomy search of the roots (sure to be always successful, unlike the fixed point methods).
If you give me some more information (I don't know in particular the degree of supervision you want : maybe you are aiming at a fully autonomous program), I can write down the Matlab program.