Using the Fundamental Theorem of Calculus

Oct 13, 2025 by magicalceylontours.com 119 views

In calculus, the Fundamental Theorem of Calculus (FTC) establishes a deep connection between derivatives and integrals. The FTC states that differentiation and integration are inverse processes, meaning that they "undo" each other. In this article, we will explore this relationship and use it to solve a problem involving the evaluation of definite integrals.

The Problem

We are given that $F^{\prime}(x)=f(x)$ and that the values of $F(x)$ at three different points are known: $F(2)=3$ , $F(1)=-1$ , and $F(0)=2$ . We are asked to find the value of $\int_0^1 f(x) \, dx - \int_1^2 f(x) \, dx$ .

Using the Fundamental Theorem of Calculus

The FTC tells us that the derivative of an antiderivative of a function is equal to the original function. In other words, if $F(x)$ is an antiderivative of $f(x)$ , then $F^{\prime}(x)=f(x)$ . We can use this fact to rewrite the given expression in terms of the antiderivative $F(x)$ .

Rewriting the Expression

We can rewrite the expression $\int_0^1 f(x) \, dx - \int_1^2 f(x) \, dx$ as follows:

\int_0^1 f(x) \, dx - \int_1^2 f(x) \, dx = \left[ F(1) - F(0) \right] - \left[ F(2) - F(1) \right]

Evaluating the Expression

Now we can substitute the known values of $F(x)$ into the expression:

\left[ F(1) - F(0) \right] - \left[ F(2) - F(1) \right] = \left[ (-1) - 2 \right] - \left[ 3 - (-1) \right]

Simplifying the expression, we get:

\left[ (-1) - 2 \right] - \left[ 3 - (-1) \right] = -3 - 4 = -7

Conclusion

Therefore, the value of $\int_0^1 f(x) \, dx - \int_1^2 f(x) \, dx$ is $\boxed{-7}$ .

Key Takeaways

The Fundamental Theorem of Calculus establishes a deep connection between derivatives and integrals.
The derivative of an antiderivative of a function is equal to the original function.
We can use the FTC to rewrite an expression involving a definite integral in terms of an antiderivative.
We can evaluate an expression involving a definite integral by substituting the known values of the antiderivative.

Additional Resources

For more information on the Fundamental Theorem of Calculus and its applications, see the following resources:

Wikipedia: Fundamental Theorem of Calculus
Khan Academy: Fundamental Theorem of Calculus
MIT OpenCourseWare: Calculus
Q&A: Understanding the Relationship Between Derivatives and Integrals ====================================================================

In our previous article, we explored the relationship between derivatives and integrals using the Fundamental Theorem of Calculus (FTC). We used the FTC to rewrite an expression involving a definite integral in terms of an antiderivative and evaluated the expression to find the value of $\int_0^1 f(x) \, dx - \int_1^2 f(x) \, dx$ . In this article, we will answer some common questions related to the FTC and its applications.

Q: What is the Fundamental Theorem of Calculus?

A: The Fundamental Theorem of Calculus (FTC) is a theorem in calculus that establishes a deep connection between derivatives and integrals. It states that differentiation and integration are inverse processes, meaning that they "undo" each other.

Q: What is the relationship between derivatives and integrals?

A: The FTC tells us that the derivative of an antiderivative of a function is equal to the original function. In other words, if $F(x)$ is an antiderivative of $f(x)$ , then $F^{\prime}(x)=f(x)$ .

Q: How can I use the FTC to rewrite an expression involving a definite integral?

A: You can use the FTC to rewrite an expression involving a definite integral by substituting the antiderivative $F(x)$ into the expression. For example, if you have the expression $\int_0^1 f(x) \, dx$ , you can rewrite it as $F(1) - F(0)$ .

Q: How can I evaluate an expression involving a definite integral?

A: You can evaluate an expression involving a definite integral by substituting the known values of the antiderivative $F(x)$ into the expression. For example, if you have the expression $F(1) - F(0)$ , you can evaluate it by substituting the known values of $F(1)$ and $F(0)$ .

Q: What are some common applications of the FTC?

A: The FTC has many applications in calculus and other areas of mathematics. Some common applications include:

Evaluating definite integrals
Finding the area under curves
Finding the volume of solids
Finding the work done by a force

Q: What are some common mistakes to avoid when using the FTC?

A: Some common mistakes to avoid when using the FTC include:

Not checking the domain of the antiderivative
Not checking the range of the antiderivative
Not using the correct antiderivative
Not evaluating the expression correctly

Q: Where can I find more information on the FTC and its applications?

A: You can find more information on the FTC and its applications in the following resources:

Conclusion

In this article, we answered some common questions related to the Fundamental Theorem of Calculus and its applications. We hope that this article has been helpful in understanding the relationship between derivatives and integrals and how to use the FTC to rewrite and evaluate expressions involving definite integrals.