To calculate the surface area generated by revolving a function $f(x)$ about the x-axis, we use the following formula for surface area (SA):

$SA = 2\pi \int_{a}^{b} f(x) \sqrt{1 + \left( \frac{df}{dx} \right)^2} \, dx$

Example Function Given:

The polynomial provided: $f(x) = 0.00430x + 0.02196x^2 + 0.36528$

We can take its derivative: $f'(x) = 0.00430 + 2 \cdot 0.02196 \cdot x = 0.00430 + 0.04392x$

Then, the integrand for surface area is: $f(x) \sqrt{1 + \left( f'(x) \right)^2} = \left(0.00430x + 0.02196x^2 + 0.36528\right) \sqrt{1 + \left(0.00430 + 0.04392x\right)^2}$

Definite Integral Calculation:

We are given limits of integration: $\int_{0}^{0.38085} \left(0.00430x + 0.02196x^2 + 0.36528\right) \sqrt{1 + \left(0.00430 + 0.04392x\right)^2} \, dx$

This integral can be computed numerically to find the surface area.

Steps for Calculation:

1. Define the function $f(x) = 0.00430x + 0.02196x^2 + 0.36528$
2. Find the derivative $f'(x) = 0.00430 + 0.04392x$
3. Set up the integrand for the surface area formula.
4. Evaluate the integral numerically from $x = 0$ to $x = 0.38085$.

Applying to the Polynomial Function:

The provided polynomial function: $f(x) = -0.0111573898143 x^{12} + 0.1703013752625 x^{11} - 1.0531878972711 x^{10} + 3.1478135332496 x^{9} - 3.0298776084592 x^{8} - 9.6898781492810 x^{7} + 40.2040231489677 x^{6} - 68.9937226700263 x^{5} + 67.439770070160 x^{4} - 39.0477196233909 x^{3} + 13.2768195135906 x^{2} + 0.6362182975726 x + C$

To calculate the surface area for this polynomial:

1. Compute the derivative $f'(x)$.
2. Set up the integral for $2\pi \int_{a}^{b} f(x) \sqrt{1 + (f'(x))^2} \, dx$.
3. Evaluate the integral over the desired range.

Surface Area of a Sphere:

For a sphere of radius $r$, the surface area (SA) is given by: $SA = 4\pi r^2$

Next Steps and Clarifications Needed:

1. What are the limits of integration for your polynomial function?
2. Would you like a step-by-step derivation for the polynomial function’s surface area?
3. Are there any specific conditions or additional details for your polynomial function?

Would you like details or have any questions?

Five Questions You Might Ask Next:

1. Can you show the step-by-step process of differentiating the polynomial function?
2. How do you numerically evaluate the integral for surface area?
3. What are the common methods for solving definite integrals?
4. How is the surface area formula derived for a function revolved around the x-axis?
5. Can you explain the significance of each term in the surface area integrand?

Tip:

When evaluating integrals for complex functions, numerical methods such as Simpson's rule or the trapezoidal rule can be very effective for obtaining approximate solutions.