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.