To find the area of the region bounded by the functions $f(x) = 9 - 1 + x$ and $g(x) = x - 9$ on the interval $[4, 10]$, we will follow these steps:

Step 1: Simplify the functions

The first function is given as: $f(x) = 9 - 1 + x = 8 + x$ The second function is: $g(x) = x - 9$

Step 2: Determine the area between the curves

The area between two curves on a given interval $[a, b]$ is given by the formula: $A = \int_a^b \left( f(x) - g(x) \right) dx$ So the area becomes: $A = \int_4^{10} \left( (8 + x) - (x - 9) \right) dx$

Step 3: Simplify the integrand

Simplify the expression inside the integral: $(8 + x) - (x - 9) = 8 + x - x + 9 = 17$ So the area integral becomes: $A = \int_4^{10} 17 \, dx$

Step 4: Integrate

The integral of a constant is: $A = 17x \Big|_4^{10}$ Now, evaluate the definite integral: $A = 17(10) - 17(4) = 170 - 68 = 102$

Final Answer:

The area of the region is: $A = 102 \, \text{units}^2$

Would you like any further details or have any questions?

Here are 5 related questions to expand on this topic:

1. How does the intersection of two curves affect the area calculation?
2. What is the geometric meaning of the integral when calculating the area between two curves?
3. How do you find the points of intersection between two functions?
4. Can the order of functions (top minus bottom) in the integral change the result? Why or why not?
5. How would the result change if the interval were different?

Tip: Always visualize the curves to ensure you're calculating the correct region's area.