Find The Definite Integral Using The Fundamental Theorem. Note: Enter The Exact Answer.$\[ \int_0^8 \left(\frac{x^3}{3} + 2x\right) \, Dx = \square \\]

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Introduction to the Fundamental Theorem of Calculus

The Fundamental Theorem of Calculus (FTC) is a fundamental concept in calculus that establishes a deep connection between the derivative and the definite integral of a function. The theorem states that differentiation and integration are inverse processes, and it provides a powerful tool for evaluating definite integrals. In this article, we will explore how to use the Fundamental Theorem of Calculus to find the definite integral of a given function.

Understanding the Fundamental Theorem of Calculus

The Fundamental Theorem of Calculus consists of two parts: the first part relates the derivative of an antiderivative to the original function, while the second part relates the definite integral of a function to its antiderivative. The first part of the theorem states that if F(x) is an antiderivative of f(x), then the derivative of F(x) is equal to f(x). Mathematically, this can be expressed as:

F'(x) = f(x)

The second part of the theorem states that the definite integral of a function f(x) from a to b can be evaluated as the antiderivative of f(x) evaluated at b, minus the antiderivative of f(x) evaluated at a. Mathematically, this can be expressed as:

∫[a, b] f(x) dx = F(b) - F(a)

Evaluating the Definite Integral Using the Fundamental Theorem

To evaluate the definite integral of a function using the Fundamental Theorem of Calculus, we need to find the antiderivative of the function and then apply the second part of the theorem. Let's consider the given function:

f(x) = (x^3)/3 + 2x

We need to find the definite integral of this function from 0 to 8.

Finding the Antiderivative

To find the antiderivative of the function f(x), we need to integrate the function with respect to x. Using the power rule of integration, we can write:

∫f(x) dx = ∫((x^3)/3 + 2x) dx

= (x^4)/12 + x^2 + C

where C is the constant of integration.

Applying the Fundamental Theorem of Calculus

Now that we have found the antiderivative of the function f(x), we can apply the second part of the Fundamental Theorem of Calculus to evaluate the definite integral. We need to evaluate the antiderivative at the upper limit of integration (x = 8) and subtract the value of the antiderivative at the lower limit of integration (x = 0).

F(8) = (8^4)/12 + 8^2 = 512/12 + 64 = 42.67 + 64 = 106.67

F(0) = (0^4)/12 + 0^2 = 0 + 0 = 0

Evaluating the Definite Integral

Now that we have evaluated the antiderivative at the upper and lower limits of integration, we can apply the second part of the Fundamental Theorem of Calculus to evaluate the definite integral.

∫[0, 8] f(x) dx = F(8) - F(0) = 106.67 - 0 = 106.67

Conclusion

In this article, we have used the Fundamental Theorem of Calculus to evaluate the definite integral of a given function. We have found the antiderivative of the function and then applied the second part of the theorem to evaluate the definite integral. The result is a precise and accurate value for the definite integral.

Frequently Asked Questions

  • What is the Fundamental Theorem of Calculus? The Fundamental Theorem of Calculus is a fundamental concept in calculus that establishes a deep connection between the derivative and the definite integral of a function.
  • How do I use the Fundamental Theorem of Calculus to evaluate a definite integral? To use the Fundamental Theorem of Calculus to evaluate a definite integral, you need to find the antiderivative of the function and then apply the second part of the theorem.
  • What is the antiderivative of a function? The antiderivative of a function is a function that, when differentiated, gives the original function.

Final Answer

The final answer is: 106.67\boxed{106.67}

Introduction

In our previous article, we explored how to use the Fundamental Theorem of Calculus to find the definite integral of a given function. We discussed the concept of the Fundamental Theorem of Calculus, how to find the antiderivative of a function, and how to apply the second part of the theorem to evaluate the definite integral. In this article, we will answer some frequently asked questions related to the Fundamental Theorem of Calculus and definite integrals.

Q&A

Q: What is the Fundamental Theorem of Calculus?

A: The Fundamental Theorem of Calculus is a fundamental concept in calculus that establishes a deep connection between the derivative and the definite integral of a function. It states that differentiation and integration are inverse processes.

Q: How do I use the Fundamental Theorem of Calculus to evaluate a definite integral?

A: To use the Fundamental Theorem of Calculus to evaluate a definite integral, you need to find the antiderivative of the function and then apply the second part of the theorem. The second part of the theorem states that the definite integral of a function f(x) from a to b can be evaluated as the antiderivative of f(x) evaluated at b, minus the antiderivative of f(x) evaluated at a.

Q: What is the antiderivative of a function?

A: The antiderivative of a function is a function that, when differentiated, gives the original function. In other words, if F(x) is an antiderivative of f(x), then F'(x) = f(x).

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

A: To find the antiderivative of a function, you need to integrate the function with respect to x. This can be done using various integration techniques, such as the power rule of integration, substitution method, and integration by parts.

Q: What is the difference between the definite integral and the indefinite integral?

A: The definite integral is a specific value that represents the area under a curve between two points, while the indefinite integral is a function that represents the family of all possible antiderivatives of a function.

Q: Can I use the Fundamental Theorem of Calculus to evaluate an improper integral?

A: No, the Fundamental Theorem of Calculus is only applicable to proper integrals. Improper integrals require a different approach, such as the limit definition of an improper integral.

Q: How do I apply the Fundamental Theorem of Calculus to evaluate a definite integral with a constant multiple?

A: When evaluating a definite integral with a constant multiple, you need to multiply the antiderivative by the constant and then apply the second part of the theorem.

Q: Can I use the Fundamental Theorem of Calculus to evaluate a definite integral with a trigonometric function?

A: Yes, the Fundamental Theorem of Calculus can be used to evaluate definite integrals with trigonometric functions. However, you need to find the antiderivative of the trigonometric function and then apply the second part of the theorem.

Conclusion

In this article, we have answered some frequently asked questions related to the Fundamental Theorem of Calculus and definite integrals. We hope that this Q&A article has provided you with a better understanding of the concept and how to apply it to evaluate definite integrals.

Frequently Asked Questions

  • What is the Fundamental Theorem of Calculus?
  • How do I use the Fundamental Theorem of Calculus to evaluate a definite integral?
  • What is the antiderivative of a function?
  • How do I find the antiderivative of a function?
  • What is the difference between the definite integral and the indefinite integral?
  • Can I use the Fundamental Theorem of Calculus to evaluate an improper integral?
  • How do I apply the Fundamental Theorem of Calculus to evaluate a definite integral with a constant multiple?
  • Can I use the Fundamental Theorem of Calculus to evaluate a definite integral with a trigonometric function?

Final Answer

The final answer is: 106.67\boxed{106.67}