Answer
The producer surplus is $\$166,666.67$
Work Step by Step
We can find the number of units $X$:
$p(x) = 125+0.002x^2$
$125+0.002X^2 = 625$
$0.002X^2 = 500$
$X^2 = 250,000$
$X = 500$
We can find the producer surplus:
$\int_{0}^{X}[P-p(x)]~dx$
$=\int_{0}^{500}[625-(125+0.002x^2)]~dx$
$=\int_{0}^{500}(500-0.002x^2)~dx$
$=500x-\frac{0.002}{3}x^3~\vert_{0}^{500}$
$= 500(500)-\frac{0.002}{3}(500)^3 - 0$
$= \$166,666.67$
The producer surplus is $\$166,666.67$
