Definite Integrals and the Fundamental Theorem of Calculus

Oct 13, 2025 by windailys.com 25 views

Introduction

In calculus, definite integrals are used to find the area under curves and can be evaluated using various techniques. Given a continuous function $f(x)$ , we can use the fundamental theorem of calculus to evaluate definite integrals. In this article, we will explore how to evaluate definite integrals using the given function $f(x)$ and its definite integral values.

Definite Integrals and the Fundamental Theorem of Calculus

The fundamental theorem of calculus states that if $f(x)$ is a continuous function on the interval $[a, b]$ , then the definite integral of $f(x)$ from $a$ to $b$ is equal to $F(b) - F(a)$ , where $F(x)$ is the antiderivative of $f(x)$ . This theorem allows us to evaluate definite integrals by finding the antiderivative of the function and then applying the limits of integration.

Given Definite Integral Values

We are given the following definite integral values:

\int_0^3 f(x) \, dx = -4 \quad \text{and} \quad \int_3^7 f(x) \, dx = 2

These values represent the area under the curve of $f(x)$ between the specified limits of integration.

Evaluating the Definite Integral

To evaluate the definite integral $\int_3^0 f(x) \, dx$ , we can use the given definite integral values and the fundamental theorem of calculus. We can rewrite the integral as:

\int_3^0 f(x) \, dx = \int_3^0 f(x) \, dx + \int_0^3 f(x) \, dx

Using the fundamental theorem of calculus, we can rewrite the integral as:

\int_3^0 f(x) \, dx = F(0) - F(3) + F(3) - F(0)

where $F(x)$ is the antiderivative of $f(x)$ .

Simplifying the Expression

We can simplify the expression by canceling out the terms:

\int_3^0 f(x) \, dx = F(0) - F(0)

This simplifies to:

\int_3^0 f(x) \, dx = 0

Evaluating the Definite Integral

To evaluate the definite integral $\int_3^7 f(x) \, dx$ , we can use the given definite integral values and the fundamental theorem of calculus. We can rewrite the integral as:

\int_3^7 f(x) \, dx = \int_3^7 f(x) \, dx + \int_0^3 f(x) \, dx

Using the fundamental theorem of calculus, we can rewrite the integral as:

\int_3^7 f(x) \, dx = F(7) - F(3) + F(3) - F(0)

where $F(x)$ is the antiderivative of $f(x)$ .

Simplifying the Expression

We can simplify the expression by canceling out the terms:

\int_3^7 f(x) \, dx = F(7) - F(0)

This simplifies to:

\int_3^7 f(x) \, dx = 2

Conclusion

In this article, we have evaluated the definite integrals $\int_3^0 f(x) \, dx$ and $\int_3^7 f(x) \, dx$ using the given definite integral values and the fundamental theorem of calculus. We have shown that the definite integral $\int_3^0 f(x) \, dx$ is equal to 0 and the definite integral $\int_3^7 f(x) \, dx$ is equal to 2.

Discussion

The definite integral is a fundamental concept in calculus that allows us to find the area under curves. In this article, we have used the fundamental theorem of calculus to evaluate definite integrals. The fundamental theorem of calculus states that if $f(x)$ is a continuous function on the interval $[a, b]$ , then the definite integral of $f(x)$ from $a$ to $b$ is equal to $F(b) - F(a)$ , where $F(x)$ is the antiderivative of $f(x)$ .

Applications

The definite integral has many applications in physics, engineering, and economics. For example, it can be used to find the area under curves, the volume of solids, and the work done by a force. It can also be used to solve problems involving motion, energy, and momentum.

Future Work

In the future, we can explore more advanced techniques for evaluating definite integrals, such as integration by parts and integration by partial fractions. We can also apply the definite integral to solve problems in physics, engineering, and economics.

References

[1] "Calculus" by Michael Spivak
[2] "Calculus: Early Transcendentals" by James Stewart
[3] "Calculus: Single Variable" by David Guichard

Appendix

The following is a list of the definite integral values used in this article:

Interval	Definite Integral Value
$[0, 3]$	$-4$
$[3, 7]$	$2$

Note: The definite integral values are given in the problem statement.

Introduction

In our previous article, we explored how to evaluate definite integrals using the fundamental theorem of calculus. We used the given definite integral values to find the area under the curve of $f(x)$ between the specified limits of integration. In this article, we will answer some frequently asked questions about evaluating definite integrals.

Q&A

Q1: What is the fundamental theorem of calculus?

A1: The fundamental theorem of calculus states that if $f(x)$ is a continuous function on the interval $[a, b]$ , then the definite integral of $f(x)$ from $a$ to $b$ is equal to $F(b) - F(a)$ , where $F(x)$ is the antiderivative of $f(x)$ .

Q2: How do I evaluate a definite integral?

A2: To evaluate a definite integral, you can use the fundamental theorem of calculus. First, find the antiderivative of the function, and then apply the limits of integration.

Q3: What is the difference between a definite integral and an indefinite integral?

A3: A definite integral is used to find the area under a curve between specific limits of integration, while an indefinite integral is used to find the antiderivative of a function.

Q4: Can I use the fundamental theorem of calculus to evaluate any definite integral?

A4: No, the fundamental theorem of calculus can only be used to evaluate definite integrals where the function is continuous on the interval of integration.

Q5: How do I find the antiderivative of a function?

A5: To find the antiderivative of a function, you can use various techniques such as substitution, integration by parts, and integration by partial fractions.

Q6: What is the relationship between the definite integral and the area under a curve?

A6: The definite integral is used to find the area under a curve between specific limits of integration.

Q7: Can I use the definite integral to solve problems in physics, engineering, and economics?

A7: Yes, the definite integral has many applications in physics, engineering, and economics. It can be used to find the area under curves, the volume of solids, and the work done by a force.

Q8: How do I apply the definite integral to solve problems in physics, engineering, and economics?

A8: To apply the definite integral to solve problems in physics, engineering, and economics, you can use the fundamental theorem of calculus and various techniques such as substitution, integration by parts, and integration by partial fractions.