Find ∫ 1 3 ( 8 X − 6 ) D X \int_1^3(8x-6) \, Dx ∫ 1 3 ( 8 X − 6 ) D X .

Mar 11, 2025 by ADMIN 74 views

**Finding the Definite Integral of a Linear Function**

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Introduction

In calculus, a definite integral is a value that represents the area under a curve between two points. In this article, we will explore how to find the definite integral of a linear function, specifically the integral of the function $8x-6$ from $x=1$ to $x=3$ . This problem is a fundamental concept in calculus and is essential for understanding more advanced topics in the field.

The Definite Integral

The definite integral of a function $f(x)$ from $a$ to $b$ is denoted as $\int_a^b f(x) \, dx$ . It represents the area under the curve of the function $f(x)$ between the points $x=a$ and $x=b$ . To find the definite integral, we need to evaluate the antiderivative of the function at the upper limit of integration and subtract the antiderivative evaluated at the lower limit of integration.

Finding the Antiderivative

To find the definite integral of the function $8x-6$ , we need to find its antiderivative. The antiderivative of a function $f(x)$ is a function $F(x)$ such that the derivative of $F(x)$ is equal to $f(x)$ . In other words, $F'(x) = f(x)$ . To find the antiderivative of $8x-6$ , we can use the power rule of integration, which states that the antiderivative of $x^n$ is $\frac{x^{n+1}}{n+1}$ .

Applying the Power Rule

Using the power rule of integration, we can find the antiderivative of $8x-6$ as follows:

\int (8x-6) \, dx = \int 8x \, dx - \int 6 \, dx

= 8 \int x \, dx - 6 \int 1 \, dx

= 8 \cdot \frac{x^2}{2} - 6x + C

= 4x^2 - 6x + C

where $C$ is the constant of integration.

Evaluating the Definite Integral

Now that we have found the antiderivative of $8x-6$ , we can evaluate the definite integral as follows:

\int_1^3 (8x-6) \, dx = \left[ 4x^2 - 6x \right]_1^3

= (4 \cdot 3^2 - 6 \cdot 3) - (4 \cdot 1^2 - 6 \cdot 1)

= (36 - 18) - (4 - 6)

= 18 + 2

= 20

Therefore, the definite integral of the function $8x-6$ from $x=1$ to $x=3$ is equal to $20$ .

Conclusion

In this article, we have explored how to find the definite integral of a linear function, specifically the integral of the function $8x-6$ from $x=1$ to $x=3$ . We have used the power rule of integration to find the antiderivative of the function and then evaluated the definite integral by applying the fundamental theorem of calculus. This problem is a fundamental concept in calculus and is essential for understanding more advanced topics in the field.

Future Directions

In future articles, we will explore more advanced topics in calculus, including the definite integral of trigonometric functions, the definite integral of exponential functions, and the definite integral of logarithmic functions. We will also explore the applications of calculus 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

Glossary

Antiderivative: A function $F(x)$ such that the derivative of $F(x)$ is equal to $f(x)$ .
Definite Integral: A value that represents the area under a curve between two points.
Fundamental Theorem of Calculus: A theorem that states that the definite integral of a function can be evaluated by applying the antiderivative of the function.
Power Rule of Integration: A rule that states that the antiderivative of $x^n$ is $\frac{x^{n+1}}{n+1}$ .

Additional Resources

[1] Khan Academy: Calculus
[2] MIT OpenCourseWare: Calculus
[3] Wolfram Alpha: Calculus

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Q: What is a definite integral?

A: A definite integral is a value that represents the area under a curve between two points. It is denoted as $\int_a^b f(x) \, dx$ and is used to find the area between a curve and the x-axis.

Q: How do I find the definite integral of a function?

A: To find the definite integral of a function, you need to find its antiderivative and then evaluate it at the upper limit of integration and subtract the antiderivative evaluated at the lower limit of integration.

Q: What is the power rule of integration?

A: The power rule of integration is a rule that states that the antiderivative of $x^n$ is $\frac{x^{n+1}}{n+1}$ . This rule is used to find the antiderivative of a function.

Q: How do I apply the power rule of integration?

A: To apply the power rule of integration, you need to identify the power of the variable in the function and then use the formula $\frac{x^{n+1}}{n+1}$ to find the antiderivative.

Q: What is the fundamental theorem of calculus?

A: The fundamental theorem of calculus is a theorem that states that the definite integral of a function can be evaluated by applying the antiderivative of the function. This theorem is used to find the definite integral of a function.

Q: How do I evaluate a definite integral?

A: To evaluate a definite integral, you need to find the antiderivative of the function and then evaluate it at the upper limit of integration and subtract the antiderivative evaluated at the lower limit of integration.

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

A: A definite integral is a value that represents the area under a curve between two points, while an indefinite integral is a function that represents the area under a curve. A definite integral is used to find the area between a curve and the x-axis, while an indefinite integral is used to find the general form of the area under a curve.

Q: How do I use the definite integral in real-world applications?

A: The definite integral is used in many real-world applications, including physics, engineering, and economics. It is used to find the area under curves, which is essential in many fields.

Q: What are some common mistakes to avoid when finding definite integrals?

A: Some common mistakes to avoid when finding definite integrals include:

Forgetting to evaluate the antiderivative at the upper and lower limits of integration
Not using the correct antiderivative
Not applying the fundamental theorem of calculus
Not checking the units of the answer

Q: How do I check my answer for a definite integral?

A: To check your answer for a definite integral, you need to:

Evaluate the antiderivative at the upper and lower limits of integration
Subtract the antiderivative evaluated at the lower limit of integration from the antiderivative evaluated at the upper limit of integration
Check the units of the answer
Check the answer against a known value or a graph of the function

Q: What are some common applications of definite integrals?

A: Some common applications of definite integrals include:

Finding the area under curves
Finding the volume of solids
Finding the work done by a force
Finding the center of mass of an object

Q: How do I use technology to find definite integrals?

A: There are many software packages and online tools that can be used to find definite integrals, including:

Wolfram Alpha
Mathematica
Maple
TI-83/84 calculators

These tools can be used to find the antiderivative of a function and to evaluate the definite integral.

Q: What are some common mistakes to avoid when using technology to find definite integrals?

A: Some common mistakes to avoid when using technology to find definite integrals include:

Not entering the correct function
Not entering the correct limits of integration
Not checking the units of the answer
Not checking the answer against a known value or a graph of the function

Q: How do I choose the right technology tool for finding definite integrals?

A: When choosing a technology tool for finding definite integrals, you should consider the following factors:

Ease of use
Accuracy
Speed
Cost
Availability of online resources and support

By considering these factors, you can choose the right technology tool for finding definite integrals.