Q $10-03.$ An $11-m$ beam is subjected to a load, and the shear force follows the equation

$V(x)=5+0\mathrm{.}25x^2$

where $V$ is the shear force and $x$ is length in distance along the beam. We know that

$V=\frac{dM}{dx}$

and $M$ is the bending moment. Integration yields the relationship

$M=M_0+{\displaystyle \underset{0}{\overset{x}{}}}Vdx$

If $M_0$ is zero and $x=11,$ calculate $M$ using $($a$)$ analytical integration and $($b$)$ trapezoidal rule. For $($b$)$ use $dx=1-m$ increments. Compute $e_t$ since the true value is known from $($a$).$ Compare results with $dx=0\mathrm{.}5,$ and $dx=0\mathrm{.}1.$

$($a$)$ What is the engineering problem? $(3)$

$($b$)$ What is the mathematical problem? $(3)$

$($c$)$ Describe a numerical method to solve this problem $(5)$

$($d$)$ Write and run Octave code to find the answer. $(12)$

$($e$)$ Write your engineering answer for a client. $(2)$