Another arc length integration example

TL;DR

Calculate the arc length of a curve between X equals one and X equals two, which is equal to 17/12.

Transcript

• [Voiceover] This right here is the graph of Y is equal to X to the third over six plus one over two X. And what I want to do in this video is to figure out the arc length along this curve between X equals one and X equals two. And so we've already highlighted that in this purple-ish color. So I encourage you to pause this video and try it out on ... Read More

Key Insights

• 🥡 The formula for arc length involves finding F prime of X, squaring it, adding one, and taking the square root.
• 😑 The definite integral of the expression gives the arc length for a specific range of X values.
• 🟰 In the provided example, the arc length between X equals one and X equals two is 17/12.
• 🫠 Calculating arc length requires knowledge of derivatives and integration techniques.
• 🫠 The arc length formula is a useful tool in various mathematical applications.
• 😑 Manipulating algebraic expressions and factoring can simplify the calculation process.
• 🫠 The arc length of a curve measures its actual length, providing insights into its shape and properties.

Questions & Answers

Q: How do you calculate arc length for a curve?

To calculate arc length, you need to find F prime of X, square it, add one, and then take the square root. This expression is then integrated over the desired range of X values to get the arc length.

Q: What is the formula for arc length for curves?

The formula for arc length for curves is ∫√(1+[F prime of X]^2) dX, where F prime of X is the derivative of the function defining the curve.

Q: How do you find F prime of X for a given function?

To find F prime of X, you need to take the derivative of the function with respect to X. In the given video, F prime of X for the function X^3/6 + 1/2X is calculated as 3X^2/6 + (-1/2X^2).

Q: What is the length of the arc in the provided example?

The length of the arc in the provided example, between X equals one and X equals two, is 17/12.

Summary & Key Takeaways

• In this video, the presenter explains how to calculate the arc length of a curve using the arc length formula.

• The presenter shows step-by-step how to find F prime of X, square it, add one, and take the square root to get the expression for arc length.

• The video concludes with the definite integral of the expression, which results in a length of 17/12 for the curve between X equals one and X equals two.