\[ C(x) = \frac{100}{1 + e^{-0.5(x - 10)}} \]  find the area under the curve between x=0 and x=20

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.

Learn how to calculate the area under the curve defined by C(x) = 100 / (1 + e^-0.5(x - 10)) from x=0 to x=20 using numerical integration. The integral of the logistic function is evaluated to find the area, which is 1000.0 square units.

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