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Magazine Chapter 6: Problem 4
Find the exact arc length of the curve over the interval. $$x=\frac{1}{3}\left(y^{2}+2\right)^{3 / 2} \text { from } y=0 \text { to } y=1$$
Short Answer
Expert verified
The exact arc length is \( \frac{4}{3} \).
Step by step solution
01
Understand Arc Length Formula
To find the arc length of a curve, we use the formula:\[ L = \int_{a}^{b} \sqrt{1 + \left(\frac{dx}{dy}\right)^2} \, dy \]where \( x = f(y) \) and \( \frac{dx}{dy} \) is the derivative of \( x \) with respect to \( y \). In this problem, \( a = 0 \) and \( b = 1 \).
02
Compute the Derivative \( \frac{dx}{dy} \)
We start by differentiating the given function:\[ x = \frac{1}{3}(y^2 + 2)^{3/2} \]Using the chain rule, the derivative \( \frac{dx}{dy} \) is:\[ \frac{dx}{dy} = \frac{1}{3} \times \frac{3}{2} \times (y^2 + 2)^{1/2} \times 2y = y(y^2 + 2)^{1/2} \]
03
Integrate to Find Arc Length
Substitute \( \frac{dx}{dy} \) into the arc length formula:\[ L = \int_{0}^{1} \sqrt{1 + (y(y^2+2)^{1/2})^2} \, dy \]Simplify the expression under the square root:\[ \sqrt{1 + y^2(y^2+2)} = \sqrt{y^4 + 2y^2 + 1} = \sqrt{(y^2+1)^2} = y^2 + 1 \]Therefore:\[ L = \int_{0}^{1} (y^2 + 1) \, dy \]
04
Solve the Integral
Find the integral:\[L = \left[ \frac{y^3}{3} + y \right]_{0}^{1} = \left(\frac{1^3}{3} + 1\right) - \left(\frac{0^3}{3} + 0\right) = \frac{1}{3} + 1 = \frac{4}{3} \]
05
State The Result
The exact arc length of the curve over the given interval is:\[ \frac{4}{3} \]
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Arc Length Formula
The arc length of a curve is a concept used to measure the distance along the curve itself from one point to another. When you need to find the arc length of a curve defined parametrically or as a function, there's a straightforward formula:
Using the formula given, we have the limits of integration from 0 to 1, and must compute the derivative \( \frac{dx}{dy} \) to substitute into the formula for further steps.
- Formula: \[ L = \int_{a}^{b} \sqrt{1 + \left(\frac{dx}{dy}\right)^2} \, dy \]
- This formula is specific to finding the arc length of a curve when you have the function in terms of \( y \), as \( x = f(y) \).
- In this formula, \( \frac{dx}{dy} \) is the derivative of \( x \) with respect to \( y \), and you integrate from \( a \) to \( b \).
- The integral helps us sum up an infinite number of tiny linear distances along the curve.
Using the formula given, we have the limits of integration from 0 to 1, and must compute the derivative \( \frac{dx}{dy} \) to substitute into the formula for further steps.
Chain Rule
To proceed with finding the arc length, you'll need to differentiate to get \( \frac{dx}{dy} \). Here comes the chain rule, a fundamental concept in calculus used to differentiate composite functions. This rule helps you understand how to take the derivative of a function of another function.
- The chain rule states: If a variable \( z \) depends on \( y \), which depends on \( x \), then \( \frac{dz}{dx} = \frac{dz}{dy} \cdot \frac{dy}{dx} \).
- For our case: \[ x = \frac{1}{3}(y^2 + 2)^{3/2} \]
- The derivative of the outer function is first differentiated: \( \frac{3}{2}(y^2 + 2)^{1/2} \).
- Afterward, you multiply by the derivative of the inside function, which is \( 2y \).
- Combine everything, and simplify to get: \[ \frac{dx}{dy} = y(y^2 + 2)^{1/2} \]
Integration
Integration is the method of calculating the integral of a function, which is essentially summing an infinite number of small areas (or lengths, in the case of arc length) under the curve. Here's how it fits into this problem:
- Once you have \( \frac{dx}{dy} \) from the chain rule:\[ \frac{dx}{dy} = y(y^2 + 2)^{1/2} \]
- Substitute into the arc length formula:\[ L = \int_{0}^{1} \sqrt{1 + (y(y^2+2)^{1/2})^2} \, dy \]
- Simplify under the square root:\[ \sqrt{1 + y^2(y^2+2)} = \sqrt{(y^2+1)^2} = y^2 + 1 \]
- You are then left with a simpler integral:\[ L = \int_{0}^{1} (y^2 + 1) \, dy \]
- Integrate term by term to get:\[ \left[ \frac{y^3}{3} + y \right]_{0}^{1} = \left(\frac{1}{3} + 1\right) - (0 + 0) = \frac{4}{3} \]
