Question: The simply supported beam is subjected to distributed forces as shown in below figure:

We want to simply the above problem by taking the integration of the distributed forces and reducing into the equivalent force loading as:

The calculation of the acting force due to distributed loading $w(x)$ can be formulated as a function:

$F_1(x)={\displaystyle \underset{0}{\overset{x}{}}}w(t)dt={\displaystyle \underset{0}{\overset{x}{}}}50e^{20t(t^2-0\mathrm{.}15)}dt,F_T=F_1(0\mathrm{.}3)$

Using the above definition, the distance $"x_f"$ can also be formulated as:

$x_f=\frac{{\displaystyle \underset{\phantom{}}{\overset{\phantom{}}{}}}t.w(t)dx}{{\displaystyle \underset{\phantom{}}{\overset{\phantom{}}{}}}w(t)dx}=\frac{F_2(x)}{F_1(x)}={\displaystyle \underset{0}{\overset{x}{}}}50e^{20t(t^2-0\mathrm{.}15)}dt={\displaystyle \underset{0}{\overset{x}{}}}50e^{20t(t^2-0\mathrm{.}15)}dt={\displaystyle \underset{0}{\overset{x}{}}}t50e^{201(t^2-0\mathrm{.}15)}dt,F_2\frac{0\mathrm{.}3}{F_1}(0\mathrm{.}3)$

Find the force $"F_T"$ and the position $"x_f"$ of the equivalent system using Taylor series expansion for $F_1(x)$ And $F_2(x)$ expanded at point $"0".$ Write all your calculations in detail $($if you use a computer program, attach the derivations to the submission, submission must be a single file$)$ at each iteration. Starting from the TS$0$ expansion, continue your iteration until approximate fractional error satisfies the ${}_s=0\mathrm{.}1$ value for integral calculations of $F_1(x)$ and $F_2(x).$ Show your final $"x_f"$ and $"F_T"$ value on the equivalent system.