这是一份southampton南安普敦大学MATH2038W1-01作业代写的成功案例
Now
$$
\mathbb{Q}^{+}=\int_{V_{1}}^{V_{2}} \Lambda_{V}\left(V, T_{2}\right) d V
$$
Furthermore
$$
W=\int_{\Gamma} P d V=\int_{T_{1}}^{T_{2}} \int_{V_{1}(T)}^{V_{2}(T)} \frac{\partial P}{\partial T} d V d T
$$
by the Gauss-Green Theorem.
$$
\int_{V_{1}}^{V_{2}} \Lambda_{V}\left(V, T_{2}\right) d V=\frac{T_{2}}{T_{2}-T_{1}} \int_{T_{1}}^{T_{2}} \int_{V_{1}(T)}^{V_{2}(T)} \frac{\partial P}{\partial T} d V d T
$$
$\operatorname{Let} T_{1} \rightarrow T_{2}=T_{}$ $$ \int_{V_{1}}^{V_{2}} \Lambda_{V}\left(V, T_{}\right) d V=T_{} \int_{V_{1}}^{V_{2}} \frac{\partial P}{\partial T}\left(V, T_{}\right) d V
$$
Divide by $V_{2}-V_{1}$ and let $V_{2} \rightarrow V_{1}=V_{*}$, to deduce
MATH2038W1-01 COURSE NOTES ：
The quantities $n ! / k !(n-k) !$ are the famous binomial coefficients, and they are denoted by
$$
\left(\begin{array}{l}
n \
k
\end{array}\right)=\frac{n !}{k !(n-k) !} \quad(n \geq 0 ; 0 \leq k \leq n)
$$
Some of their special values are
$$
\begin{gathered}
\left(\begin{array}{l}
n \
0
\end{array}\right)=1 \quad(\forall n \geq 0) ; \quad\left(\begin{array}{l}
n \
1
\end{array}\right)=n \quad(\forall n \geq 0) ; \
\left(\begin{array}{l}
n \
2
\end{array}\right)=n(n-1) / 2 \quad(\forall n \geq 0) ; \quad\left(\begin{array}{l}
n \
n
\end{array}\right)=1 \quad(\forall n \geq 0) .
\end{gathered}
$$
It is convenient to define $\left(\begin{array}{l}n \ k\end{array}\right)$ to be 0 if $k<0$ or if $k>n$.