calculate the area between -1/24(x-2)(x-4)(x-6)(x-9)+2 and 1/12(x-11/2)^2+1 from 1.45145 to 9.36174

To calculate the area between the two curves \( f(x) = -\frac{1}{24}(x-2)(x-4)(x-6)(x-9)+2 \) and \( g(x) = \frac{1}{12}(x-\frac{11}{2})^2+1 \) from \( x = 1.45145 \) to \( x = 9.36174 \), we need to find the definite integral of the difference of the two functions over the given interval.

The area \( A \) can be calculated as follows:

\[ A = \int_{1.45145}^{9.36174} \left[ g(x) - f(x) \right] \, dx \]

First, we need to express the difference of the functions:

\[ g(x) - f(x) = \frac{1}{12}\left(x-\frac{11}{2}\right)^2+1 - \left(-\frac{1}{24}(x-2)(x-4)(x-6)(x-9)+2\right) \]

Simplify the expression inside the integral:

\[ g(x) - f(x) = \frac{1}{12}\left(x-\frac{11}{2}\right)^2+1 + \frac{1}{24}(x-2)(x-4)(x-6)(x-9) - 2 \]

\[ g(x) - f(x) = \frac{1}{12}\left(x-\frac{11}{2}\right)^2 + \frac{1}{24}(x-2)(x-4)(x-6)(x-9) - 1 \]

Next, we compute this integral using a numerical method:

\[ A = \int_{1.45145}^{9.36174} \left[\frac{1}{12}\left(x-\frac{11}{2}\right)^2 + \frac{1}{24}(x-2)(x-4)(x-6)(x-9) - 1 \right] dx \]

Let's compute this integral using Python:

```python
import sympy as sp

x = sp.symbols('x')
f = -1/24 * (x-2)*(x-4)*(x-6)*(x-9) + 2
g = 1/12 * (x-11/2)**2 + 1
integrand = g - f

area = sp.integrate(integrand, (x, 1.45145, 9.36174))
area.evalf()
```

Running this computation gives us the result for the area. Let me do that for you.The area between the two curves from \( x = 1.45145 \) to \( x = 9.36174 \) is approximately \(-7.7026\). The negative value indicates that the function \( f(x) \) is generally above \( g(x) \) over the given interval, thus the area calculated is effectively the integral of \( f(x) - g(x) \).

Would you like to understand the steps in more detail or have any questions?

Here are some questions you might want to ask next:
1. How do you set up the integral for the area between two curves?
2. Can you show the detailed steps for simplifying the integrand?
3. How can you determine which function is above the other over an interval?
4. What numerical methods can be used for approximating integrals?
5. How would this area change if the interval or functions were different?

**Tip:** When computing the area between two curves, always check which function is on top over the given interval to set up the integrand correctly.

Learn how to calculate the area between two curves using definite integrals. This example involves finding the area between the polynomial functions -1/24(x-2)(x-4)(x-6)(x-9)+2 and 1/12(x-11/2)^2+1 from x = 1.45145 to x = 9.36174. Step-by-step solution and numerical method included.

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