Answer
The Consumers’ Surplus is given by:
$$
\begin{aligned}
\text {Consumers’ surplus }&=\int_{0}^{q_{0}}[D(q)- p_{0}] d q \\ &=\int_{0}^{3}\left[\frac{200}{(3 q+1)^{2}}-2\right] d q\\
&=54.
\end{aligned}
$$
Work Step by Step
the demand function is given by
$$
D(q)=\frac{200}{(3 q+1)^{2}}
$$
equilibrium supply $p_0$ is when $q=3$, so we have:
$$
\begin{aligned}
p_0=D(3)&=\frac{200}{(3 (3)+1)^{2}}\\
&=\frac{200}{(10)^{2}}\\
&=\frac{200}{(100)}\\
&=2
\end{aligned}
$$
Now, we know that:
$$
\text {Consumers’ Surplus }=\int_{0}^{q_{0}}[D(q)- p_{0}] d q
$$
So, the Consumers’ Surplus is given by:
$$
\begin{aligned}
\text {Consumers’ surplus }&\int_{0}^{q_{0}}[D(q)- p_{0}] d q \\ &=\int_{0}^{3}\left[\frac{200}{(3 q+1)^{2}}-2\right] d q\\
&=\int_{0}^{3} \frac{200}{(3 q+1)^{2}} d q-\int_{0}^{3} 2 d q \\
&\quad\quad\quad\left[\begin{array}{c}{ \text {Let }u=3q+1, q : 0 \rightarrow 3 } \\ { \text {Then }d u=3d q}, u:1 \rightarrow 10 \end{array}\right] \\
&=\frac{1}{3} \int_{1}^{10} \frac{200}{u^{2}} d u-\int_{0}^{3} 2 d q\\
&=\frac{200}{3} \int_{1}^{10} u^{-2} d u-\int_{0}^{3} 2 d q \\
&=\left.\frac{200}{3} \cdot \frac{u^{-1}}{-1}\right|_{1} ^{10}-\left.2 q\right|_{0} ^{3}\\
&=-\frac{200}{30}+\frac{200}{3}-6\\
&=54.
\end{aligned}
$$
