$($a$)$ A $5-$metre$-$long uniform beam is simply supported at both ends and is subjected to the uniformly

distributed load shown in the Figure $1.$

Figure $1$: A simply supported beam subjected to a uniformly distributed load.

The deflection of the beam is given by the following differential equation:

$\frac{d^{2}y}{d{x}^2}=-\frac{M(x)}{EI}$

where $y$ is the deflection, $x$ is the coordinate measured along the length of the beam, $M(x)$ is the bending

moment, and $EI=10000kN{m}^2$ is the flexural rigidity of the beam. The following data are obtained from

measuring the deflection of the beam versus position:

Based on the data in the table above, use the central finite$-$difference method to determine the bending

moment of the beam at the location $x=2\mathrm{.}5m.$

$($b$)$ Consider the following single definite integral ${\displaystyle \underset{0}{\overset{3}{}}}{e}^{-x^2}dx$

$($b$1)$ Transform the integral to the form ${\displaystyle \underset{-1}{\overset{1}{}}}f(t)dt$

$($b$2)$ Evaluate the value of the integral using three$-$point quadrature$($a$)$ A $5-$metre$-$long uniform beam is simply supported at both ends and is subjected to the uniformly

distributed load shown in the Figure $1.$

Figure $1$: A simply supported beam subjected to a uniformly distributed load.

The deflection of the beam is given by the following differential equation:

$\frac{d^{2}y}{d{x}^2}=-\frac{M(x)}{EI}$

where $y$ is the deflection, $x$ is the coordinate measured along the length of the beam, $M(x)$ is the bending

moment, and $EI=10000kN{m}^2$ is the flexural rigidity of the beam. The following data are obtained from

measuring the deflection of the beam versus position:

Based on the data in the table above, use the central finite$-$difference method to determine the bending

moment of the beam at the location $x=2\mathrm{.}5m.$

$($b$)$ Consider the following single definite integral ${\displaystyle \underset{0}{\overset{3}{}}}{e}^{-x^2}dx$

$($b$1)$ Transform the integral to the form ${\displaystyle \underset{-1}{\overset{1}{}}}f(t)dt$

$($b$2)$ Evaluate the value of the integral using three$-$point quadrature