Applying the Arc Length Formula

Oct 13, 2025 by windailys.com 124 views

Introduction

In mathematics, the length of a curve is a fundamental concept that has numerous applications in various fields, including physics, engineering, and computer science. The length of a curve can be calculated using the arc length formula, which is a powerful tool for determining the distance along a curve between two points. In this article, we will explore the concept of curve length and apply it to find the length of the curve $y=\frac{x^2}{2}$ , for $0 \leq x \leq 2 \sqrt{2}$ .

What is Curve Length?

Curve length, also known as arc length, is a measure of the distance along a curve between two points. It is a fundamental concept in mathematics and has numerous applications in various fields. The length of a curve can be calculated using the arc length formula, which is a mathematical expression that relates the length of a curve to its derivative.

The Arc Length Formula

The arc length formula is given by:

L = \int_{a}^{b} \sqrt{1 + \left(\frac{dy}{dx}\right)^2} dx

where $L$ is the length of the curve, $a$ and $b$ are the limits of integration, and $\frac{dy}{dx}$ is the derivative of the function.

Finding the Derivative

To find the length of the curve $y=\frac{x^2}{2}$ , we need to find the derivative of the function. The derivative of the function is given by:

\frac{dy}{dx} = \frac{d}{dx} \left(\frac{x^2}{2}\right) = x

Applying the Arc Length Formula

Now that we have the derivative of the function, we can apply the arc length formula to find the length of the curve. The arc length formula is given by:

L = \int_{a}^{b} \sqrt{1 + \left(\frac{dy}{dx}\right)^2} dx

Substituting the derivative of the function into the arc length formula, we get:

L = \int_{0}^{2 \sqrt{2}} \sqrt{1 + x^2} dx

Evaluating the Integral

To evaluate the integral, we can use the trigonometric substitution method. Let $x = \tan(\theta)$ , then $dx = \sec^2(\theta) d\theta$ . Substituting these expressions into the integral, we get:

L = \int_{0}^{\frac{\pi}{4}} \sqrt{1 + \tan^2(\theta)} \sec^2(\theta) d\theta

Simplifying the integral, we get:

L = \int_{0}^{\frac{\pi}{4}} \sec^3(\theta) d\theta

Using the Trigonometric Identity

To evaluate the integral, we can use the trigonometric identity:

\sec^3(\theta) = \sec(\theta) \cdot \sec^2(\theta)

Substituting this expression into the integral, we get:

L = \int_{0}^{\frac{\pi}{4}} \sec(\theta) \cdot \sec^2(\theta) d\theta

Evaluating the Integral

To evaluate the integral, we can use the substitution method. Let $u = \sec(\theta)$ , then $du = \sec(\theta) \tan(\theta) d\theta$ . Substituting these expressions into the integral, we get:

L = \int_{1}^{2} u^2 du

Evaluating the integral, we get:

L = \left[\frac{u^3}{3}\right]_{1}^{2}

Simplifying the expression, we get:

L = \frac{8}{3} - \frac{1}{3}

The Final Answer

The length of the curve $y=\frac{x^2}{2}$ , for $0 \leq x \leq 2 \sqrt{2}$ is:

L = \frac{7}{3}

Conclusion

$In this article, we explored the concept of curve length and applied it to find the length of the curve $y=\frac{x^2}{2}$, for $0 \leq x \leq 2 \sqrt{2}$. We used the arc length formula and the trigonometric substitution method to evaluate the integral and find the length of the curve. The final answer is $\frac{7}{3}$.<br/> **Curve Length Q&A: Frequently Asked Questions**$

Introduction

In our previous article, we explored the concept of curve length and applied it to find the length of the curve $y=\frac{x^2}{2}$ , for $0 \leq x \leq 2 \sqrt{2}$ . In this article, we will answer some frequently asked questions about curve length and provide additional information to help you better understand this concept.

Q: What is curve length?

A: Curve length, also known as arc length, is a measure of the distance along a curve between two points. It is a fundamental concept in mathematics and has numerous applications in various fields.

Q: How is curve length calculated?

A: Curve length is calculated using the arc length formula, which is given by:

L = \int_{a}^{b} \sqrt{1 + \left(\frac{dy}{dx}\right)^2} dx

where $L$ is the length of the curve, $a$ and $b$ are the limits of integration, and $\frac{dy}{dx}$ is the derivative of the function.

Q: What is the derivative of a function?

A: The derivative of a function is a measure of how the function changes as its input changes. It is denoted by $\frac{dy}{dx}$ and is calculated using the limit definition:

\frac{dy}{dx} = \lim_{h \to 0} \frac{f(x+h) - f(x)}{h}

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

A: There are several methods for finding the derivative of a function, including:

Using the power rule: If $f(x) = x^n$ , then $f'(x) = nx^{n-1}$
Using the product rule: If $f(x) = u(x)v(x)$ , then $f'(x) = u'(x)v(x) + u(x)v'(x)$
Using the quotient rule: If $f(x) = \frac{u(x)}{v(x)}$ , then $f'(x) = \frac{u'(x)v(x) - u(x)v'(x)}{v(x)^2}$

Q: What is the arc length formula?

A: The arc length formula is given by:

L = \int_{a}^{b} \sqrt{1 + \left(\frac{dy}{dx}\right)^2} dx

where $L$ is the length of the curve, $a$ and $b$ are the limits of integration, and $\frac{dy}{dx}$ is the derivative of the function.

Q: How do I evaluate the integral in the arc length formula?

A: There are several methods for evaluating the integral in the arc length formula, including:

Using the substitution method: Let $u = \sqrt{1 + \left(\frac{dy}{dx}\right)^2}$ , then $du = \frac{dy}{dx} \frac{d}{dx} \left(\sqrt{1 + \left(\frac{dy}{dx}\right)^2}\right) dx$
Using the trigonometric substitution method: Let $x = \tan(\theta)$ , then $dx = \sec^2(\theta) d\theta$

Q: What is the final answer for the length of the curve $y=\frac{x^2}{2}$ , for $0 \leq x \leq 2 \sqrt{2}$ ?

A: The final answer for the length of the curve $y=\frac{x^2}{2}$ , for $0 \leq x \leq 2 \sqrt{2}$ is:

L = \frac{7}{3}

Conclusion

In this article, we answered some frequently asked questions about curve length and provided additional information to help you better understand this concept. We hope that this article has been helpful in clarifying any confusion you may have had about curve length. If you have any further questions, please don't hesitate to ask.