To find the area under the curve defined by the function $C(x) = \frac{100}{1 + e^{-0.5(x - 10)}}$ between $x = 0$ and $x = 20$, we need to compute the definite integral:

$\text{Area} = \int_{0}^{20} C(x) \, dx = \int_{0}^{20} \frac{100}{1 + e^{-0.5(x - 10)}} \, dx$

This integral doesn't have a simple analytical solution, so we typically evaluate it numerically. I'll calculate this for you now.The area under the curve $C(x) = \frac{100}{1 + e^{-0.5(x - 10)}}$ between $x = 0$ and $x = 20$ is 1000.0 square units.

Would you like more details or have any questions?

Here are some related questions you might find interesting:

1. What is the value of the integral from $x = 0$ to $x = 10$?
2. How does the function $C(x)$ behave as $x$ approaches infinity?
3. What is the derivative of $C(x)$, and what does it represent?
4. How would the area change if the range was from $x = 5$ to $x = 15$?
5. How does varying the parameter in the exponent (e.g., from $-0.5$ to $-1$) affect the shape of the curve?

Tip: When dealing with complex integrals that lack simple analytical solutions, numerical methods like the trapezoidal rule or Simpson's rule can provide accurate approximations.