Given a function f (x), that is continuous on [a, b],

i) What fundamental theorem gives you an easier way to calculate definite integrals?

The **Fundamental Theorem of Calculus** provides an easier way to calculate definite integrals. The theorem consists of two parts, and the part that gives us an easier way to compute definite integrals can be stated as follows:

### Fundamental Theorem of Calculus (Part 2)
If \( f(x) \) is continuous on the interval \([a, b]\) and \( F(x) \) is an antiderivative of \( f(x) \) on \([a, b]\), then:
\[ \int_{a}^{b} f(x) \, dx = F(b) - F(a) \]

### Explanation:
This means that to find the definite integral of a continuous function \( f(x) \) over \([a, b]\), you need to:
1. Find an antiderivative \( F(x) \) of \( f(x) \).
2. Evaluate \( F(x) \) at the endpoints \( a \) and \( b \).
3. Subtract \( F(a) \) from \( F(b) \) to get the value of the definite integral.

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

Here are 8 related questions to expand your understanding:

1. What is an antiderivative, and how do you find it for a given function?
2. How can you verify if a function is continuous on a given interval?
3. What is the first part of the Fundamental Theorem of Calculus?
4. How do you handle definite integrals where the function has discontinuities?
5. Can the Fundamental Theorem of Calculus be applied to improper integrals?
6. How does the Fundamental Theorem of Calculus relate to the concept of accumulation functions?
7. What are some common techniques for finding antiderivatives of complex functions?
8. How does the Fundamental Theorem of Calculus simplify the process of finding the area under a curve?

### Tip:
Always ensure that the function \( f(x) \) is continuous on the interval \([a, b]\) before applying the Fundamental Theorem of Calculus to avoid incorrect results.

Explore how the Fundamental Theorem of Calculus simplifies the calculation of definite integrals. Understand the process of finding antiderivatives and applying them to evaluate integrals over given intervals. Suitable for advanced high school and college students.

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