What property do we also need f (x) to have on [a, b], to use the theorem from (i) to calculate

areas?

To use the Fundamental Theorem of Calculus to calculate areas under the curve \( f(x) \) on the interval \([a, b]\), the function \( f(x) \) must be **non-negative** on \([a, b]\). This ensures that the integral represents the actual geometric area under the curve and above the \( x \)-axis.

### Explanation:
While the Fundamental Theorem of Calculus allows us to calculate the definite integral of \( f(x) \) over \([a, b]\), if \( f(x) \) is not non-negative, the integral might yield a value that represents the net area (taking into account the areas above and below the \( x \)-axis) rather than the total geometric area. To ensure that the result of the integral corresponds to the geometric area, \( f(x) \) should satisfy:

\[ f(x) \geq 0 \text{ for all } x \in [a, b] \]

### Summary:
So, to use the Fundamental Theorem of Calculus for calculating areas under the curve, \( f(x) \) must be both continuous and non-negative on the interval \([a, b]\).

Would you like more details or have any questions about this?

Here are 8 related questions to expand your understanding:

1. What happens if \( f(x) \) is negative on some parts of the interval \([a, b]\)?
2. How can you calculate the area between two curves using definite integrals?
3. How do you handle functions that are not non-negative on the entire interval?
4. What is the geometric interpretation of the definite integral if \( f(x) \) is not non-negative?
5. How can you split the interval \([a, b]\) if \( f(x) \) changes sign within it?
6. What methods can be used to ensure that the integral calculation accounts for non-negative values only?
7. How do you find the antiderivative of a piecewise function?
8. What are some real-world applications of calculating areas under curves using integrals?

### Tip:
When dealing with functions that are not non-negative on the entire interval, consider breaking the interval into subintervals where the function is non-negative and then summing the absolute values of the integrals over these subintervals to find the total area.

Learn how to use the Fundamental Theorem of Calculus to calculate areas under the curve ( f(x) ) on the interval [a, b]. Ensure f(x) is non-negative for accurate area calculations. Understand the implications of negative values of f(x) on the interval. Suitable for advanced high school students and calculus enthusiasts.

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