g\left(x\right)=-0.03\left(x-4.3\right)^{4}+9 and f\left(x\right)=-0.0274+3.58x+-0.625x^{2}+0.0251x^{3}+8.6\cdot10^{-5}\cdot x^{4} area between the two

To find the **area between** the two functions:

\[
g(x) = -0.03(x - 4.3)^4 + 9
\]

\[
f(x) = -0.0274 + 3.58x - 0.625x^2 + 0.0251x^3 + 8.6 \times 10^{-5} x^4
\]

we follow these steps:

### **1. Find Intersection Points**
The area is computed as:

\[
A = \int_{a}^{b} |f(x) - g(x)| \,dx
\]

where \( a \) and \( b \) are the **x-values where** \( f(x) = g(x) \). We first need to solve:

\[
-0.03(x - 4.3)^4 + 9 = -0.0274 + 3.58x - 0.625x^2 + 0.0251x^3 + 8.6 \times 10^{-5} x^4
\]

### **2. Compute the Definite Integral**
Once we have the limits of integration \( a \) and \( b \), we integrate:

\[
A = \int_{a}^{b} |f(x) - g(x)| \,dx
\]

I'll now calculate the intersection points and evaluate the integral. numpy as np
import sympy as sp
from scipy.integrate import quad

# Define x as a symbol
x = sp.Symbol('x')

# Define the functions
g_x = -0.03*(x - 4.3)**4 + 9
f_x = -0.0274 + 3.58*x - 0.625*x**2 + 0.0251*x**3 + 8.6e-5*x**4

# Solve for intersection points f(x) = g(x)
intersection_points = sp.solveset(f_x - g_x, x, domain=sp.S.Reals)
intersection_pointsThe functions intersect at approximately:  
\[
x \approx 0.229 \quad \text{and} \quad x \approx 8.289
\]
Now, I'll compute the definite integral:

\[
A = \int_{0.229}^{8.289} |f(x) - g(x)| \,dx
\] Convert functions to numerical form for integration
f_numeric = sp.lambdify(x, f_x, 'numpy')
g_numeric = sp.lambdify(x, g_x, 'numpy')

# Define absolute difference function
abs_diff = lambda x: abs(f_numeric(x) - g_numeric(x))

# Compute the definite integral
a, b = 0.229, 8.289
area, error = quad(abs_diff, a, b)
areaThe area between the two curves from \( x \approx 0.229 \) to \( x \approx 8.289 \) is approximately **25.45** square units.

Would you like a step-by-step breakdown of the integration process?

### Related Questions:
1. How do you determine the points of intersection algebraically?
2. What happens if the curves cross more than twice?
3. How can we approximate the integral numerically if a function is difficult to integrate?
4. How does the shape of \( g(x) \) (a quartic function) affect the area calculation?
5. How can this method be applied to real-world problems, such as physics or economics?

**Tip:** When finding the area between curves, always check if the functions cross multiple times to correctly handle absolute values in integration.

Find the area between the functions g(x) = -0.03(x - 4.3)^4 + 9 and f(x) = -0.0274 + 3.58x - 0.625x^2 + 0.0251x^3 + 8.6 × 10^{-5} x^4.

Learn how to compute the area between two functions using definite integration. This step-by-step guide walks through solving the intersection points, setting up the integral, and evaluating the absolute difference between f(x) and g(x). Suitable for students studying calculus.

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