Answer
The Consumers’ Surplus is given by:
$$
\begin{aligned}
\text {Consumers’ surplus }&= \int_{0}^{q_{0}}[D(q)- p_{0}] d q \\ &=\int_{0}^{6}\left[\frac{32,00}{(2 q+8)^{3}}-4\right] d q\\
&= 81
\end{aligned}
$$
Work Step by Step
the demand function is given by
$$
D(q)=\frac{32,000}{(2 q+8)^{3}},
$$
equilibrium supply $p_0$ is when $q=6$, so we have:
$$
\begin{aligned}
p_0=D(6)&=\frac{32,000}{(2 (6)+8)^{3}}\\
&=\frac{32,000}{(2 0)^{3}}\\
&=\frac{32,000}{(8,000}\\
&=4
\end{aligned}
$$
Now, we know that:
$$
\text {Consumers’ Surplus }=\int_{0}^{q_{0}}[D(q)- p_{0}] d q
$$
So, the Consumers’ Surplus is given by:
$$
\begin{aligned}
\text {Consumers’ surplus }&\int_{0}^{q_{0}}[D(q)- p_{0}] d q \\ &=\int_{0}^{6}\left[\frac{32,00}{(2 q+8)^{3}}-4\right] d q\\
&=\int_{0}^{6} \frac{32,00}{(2 q+8)^{3}} d q-\int_{0}^{6} 4 d q \\
&\quad\quad\quad\left[\begin{array}{c}{ \text {Let }u=2q+8, q : 0 \rightarrow 6 } \\ { \text {Then }d u=2d q}, u:8 \rightarrow 20 \end{array}\right] \\
=& \frac{1}{2} \int_{8}^{20} \frac{32,000}{u^{3}} d u-\int_{0}^{6} 4 d q \\=& 16,000 \int_{8}^{20} u^{-3} d u-\int_{0}^{6} 4 d q \\=& 16,\left.00 \cdot \frac{u^{-2}}{-2}\right|_{8} ^{20}-\left.4 q\right|_{0} ^{6} \\=&-\left.\frac{8000}{u^{2}}\right|_{8} ^{20}-24 \\=&-\frac{8000}{400}+\frac{8000}{64}-24 \\=& 81
\end{aligned}
$$
