$($a$)$ The beam in Figure $2$ is subjected to a couple moment at point $C,$ determine the slope at point $C$ using the method of double integration. Take $E=200$GPa,$I=400({10}^6)m{m}^4(10\%)$

$($b$)$ Under the same external loading, the top of the beam relative to the bottom of the beam has a $100C$ increase in temperature. If $=12\frac{{10}^{-6}}{}C$ and the depth of the beam is $200mm,$ determine the slope at point $C$ due to both the couple moment and the temperature gradient. $(15\%)$

$U_{\text{t}\text{e}\text{m}\text{p}}={\displaystyle \underset{0}{\overset{L}{}}}\frac{m{T}_{m}dx}{c}$

Figure $2$