Find the Consumer Surplus, given the demand and supply equations $$ D(x)=\frac{405}{\sqrt{x}} $$ $$ S(x)=5\sqrt{x} $$ The equilibrium point is $(81,45)$.
I know the formula for consumer surplus, but I am stuck on finding the integral of $405/\sqrt{x}$.
On
Point of Equilibrium for Supply and Demand
$$ D(q_e)=S(q_e) $$ $$ \frac{405}{\sqrt{q_e}}=5\sqrt{q_e} $$ $$ 405=5|q_e|$$ $$ |q_e|=\frac{405}{5}=81$$ The next step is to find the market price, which is $$ P_{mkt}(q_e)=D(q_e)=S(q_e)=5\sqrt{81}=5\cdot 9=45 $$ Therefore, the point of equilibrium is $$ (q_e, P_{mkt}(q_e))=(81, 45)$$ Consumer Surplus
By definition the consumer surplus $CS$, is the area between the demand curve and the market price from $0$ to the quantity at the point of equilibrium. Mathematically this is expressed as $$ CS= \int_{0}^{q_e} D(q) - P_{mkt}(q_e)\ dq $$ So in this case we have, $$ CS= \int_{0}^{81} \frac{405}{\sqrt{q}} - 45\ dq $$ $$= 405\int_{0}^{81} q^{-\frac{1}{2}}dq- 45\int_0^{81} dq $$ $$= 405\left[\frac{q^{\frac{1}{2}}}{\frac12}\right]_{0}^{81} - 45(81-0)$$ $$= 2(405)(\sqrt{81}-0)- 45(81-0)$$ $$= 2(405)(9)- 45(81) $$ $$=7290-3645=3645 $$
For context, I grabbed this picture from Wikipedia
The red area is the integral of $D(x) - 45$ from $0$ to $81$. Namely, $$ \int_0^{81} \left(\frac{405}{\sqrt{x}}-45\right)\,dx $$ To integrate, write $405/\sqrt{x}$ as $x^{-1/2}$ and use the formula $\int x^a = x^{a+1}/(a+1)$.
To check the answer, you can use Wolfram Alpha: it's $3645$.