Adapted from problem $6\mathrm{.}29(20$ marks$)$ Using the same data as the previous question:

a$.$ Compute analytically the modulus of resilience, $U_r.$ This relies on knowing the modulus of elasticity, $E,$ and knowing the elastic limit ${}_e,$ from parts $5.$b and $5.$c$.$

$2$

b$.$ Compute the modulus of toughess, $U_t.$ This must be done numerically, as we have no general equation for the stress$/$strain curve beyond the elastic regime. We instead use the trapezoidal rule. We take the area between two points as the area under the line that connects them: the resulting shape is a trapezoid, with a left side of ${}_1,$ a right side of ${}_2,$ and a width of ${}_2-{}_1.$ The area, then, is

$A_1=\frac{1}{2}({}_2+{}_1)({}_2-{}_1)$

The area of the whole curve is the sum of the trapezoidal areas:

$U_t={\displaystyle \underset{?}{\overset{?}{}}}A=\frac{1}{2}{\displaystyle \underset{?}{\overset{?}{}}}({}_n+{}_{n-1})({}_n-{}_{n-1})$