The Concept of Integration

Oct 13, 2025 by miodeya.com 68 views

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Introduction

In mathematics, finding the area enclosed by a curve is a fundamental problem that has numerous applications in various fields, including physics, engineering, and economics. The area enclosed by a curve can be calculated using various methods, including integration. In this article, we will explore how to find the area enclosed by the curve $y=x^2+2x$ between the ordinates $x=0$ and $x=2$ .

Understanding the Problem

The problem requires us to find the area enclosed by the curve $y=x^2+2x$ between the ordinates $x=0$ and $x=2$ . To solve this problem, we need to understand the concept of integration and how it can be used to find the area under a curve. Integration is a mathematical process that involves finding the area under a curve by summing up the areas of infinitesimally small rectangles.

The Concept of Integration

Integration is a fundamental concept in calculus that involves finding the area under a curve. The area under a curve can be calculated using the definite integral, which is denoted by the symbol $\int$ . The definite integral is used to find the area between two points on a curve. In this case, we need to find the area between the ordinates $x=0$ and $x=2$ .

Finding the Area Using Integration

To find the area enclosed by the curve $y=x^2+2x$ between the ordinates $x=0$ and $x=2$ , we need to use the definite integral. The definite integral is given by the formula:

\int_{a}^{b} f(x) dx = F(b) - F(a)

where $f(x)$ is the function that defines the curve, and $a$ and $b$ are the limits of integration.

In this case, the function that defines the curve is $y=x^2+2x$ , and the limits of integration are $x=0$ and $x=2$ . Therefore, we need to evaluate the definite integral:

\int_{0}^{2} (x^2+2x) dx

Evaluating the Definite Integral

To evaluate the definite integral, we need to find the antiderivative of the function $f(x)=x^2+2x$ . The antiderivative of $f(x)$ is given by:

F(x) = \frac{x^3}{3} + x^2

Now, we can evaluate the definite integral by substituting the limits of integration:

\int_{0}^{2} (x^2+2x) dx = F(2) - F(0)

\int_{0}^{2} (x^2+2x) dx = \left(\frac{2^3}{3} + 2^2\right) - \left(\frac{0^3}{3} + 0^2\right)

\int_{0}^{2} (x^2+2x) dx = \left(\frac{8}{3} + 4\right) - 0

\int_{0}^{2} (x^2+2x) dx = \frac{8}{3} + 4

\int_{0}^{2} (x^2+2x) dx = \frac{8}{3} + \frac{12}{3}

\int_{0}^{2} (x^2+2x) dx = \frac{20}{3}

Conclusion

In this article, we explored how to find the area enclosed by the curve $y=x^2+2x$ between the ordinates $x=0$ and $x=2$ . We used the concept of integration and the definite integral to find the area under the curve. The definite integral is a powerful tool that can be used to find the area under a curve, and it has numerous applications in various fields.

Final Answer

The final answer to the problem is $\boxed{\frac{20}{3}}$ .

References

[1] Calculus, 3rd edition, Michael Spivak
[2] Calculus, 2nd edition, James Stewart
[3] Integration, 1st edition, David Guichard

Glossary

Definite Integral: A mathematical concept that involves finding the area under a curve between two points.
Antiderivative: A function that is the inverse of the derivative of another function.
Limits of Integration: The points on a curve between which the area is to be found.
Area Under a Curve: The area enclosed by a curve between two points.

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Introduction

In our previous article, we explored how to find the area enclosed by the curve $y=x^2+2x$ between the ordinates $x=0$ and $x=2$ . In this article, we will answer some frequently asked questions related to finding the area enclosed by a curve.

Q: What is the concept of integration?

A: Integration is a mathematical process that involves finding the area under a curve by summing up the areas of infinitesimally small rectangles. It is a fundamental concept in calculus that has numerous applications in various fields.

Q: How do I find the area under a curve?

A: To find the area under a curve, you need to use the definite integral. The definite integral is given by the formula:

\int_{a}^{b} f(x) dx = F(b) - F(a)

where $f(x)$ is the function that defines the curve, and $a$ and $b$ are the limits of integration.

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

A: A definite integral is used to find the area between two points on a curve, while an indefinite integral is used to find the antiderivative 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 that defines the curve. The antiderivative is given by:

F(x) = \int f(x) dx

Then, you need to substitute the limits of integration into the antiderivative to find the area under the curve.

Q: What is the significance of the limits of integration?

A: The limits of integration are the points on a curve between which the area is to be found. They are essential in defining the area under the curve.

Q: Can I use integration to find the area under a curve with a discontinuity?

A: Yes, you can use integration to find the area under a curve with a discontinuity. However, you need to be careful when evaluating the definite integral, as the antiderivative may not be defined at the point of discontinuity.

Q: How do I apply integration in real-world problems?

A: Integration has numerous applications in various fields, including physics, engineering, and economics. Some examples of real-world problems that involve integration include:

Finding the area under a curve to determine the volume of a solid
Finding the area under a curve to determine the work done by a force
Finding the area under a curve to determine the value of a definite integral