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Magazine Chapter 5: Problem 49
Find the length of the curve \(y=\frac{1}{2} x^{3 / 2}\) between \(x=0\) and \(x=1\).
Short Answer
Expert verified
The length of the curve is approximately 1.479 units.
Step by step solution
01
Understand the Problem
We are tasked with finding the length of a curve defined by the function \( y = \frac{1}{2} x^{3/2} \) between \( x = 0 \) and \( x = 1 \). This involves using the formula for the arc length of a curve over a specific interval.
02
Recall the Arc Length Formula
To find the length of a curve \( y = f(x) \) from \( x = a \) to \( x = b \), use the formula: \[ L = \int_{a}^{b} \sqrt{1 + \left(\frac{dy}{dx}\right)^2} \, dx. \]
03
Differentiate the Function
Find the derivative of \( y \) with respect to \( x \). Given \( y = \frac{1}{2} x^{3/2} \), we use the power rule: \[\frac{dy}{dx} = \frac{3}{4} x^{1/2}.\]
04
Substitute and Simplify the Expression
Substitute \( \frac{dy}{dx} = \frac{3}{4} x^{1/2} \) into the arc length formula:\[L = \int_{0}^{1} \sqrt{1 + \left(\frac{3}{4} x^{1/2}\right)^2} \, dx.\]Simplify this to \[L = \int_{0}^{1} \sqrt{1 + \frac{9}{16} x} \, dx.\]
05
Evaluate the Integral
We then integrate this expression from 0 to 1. This can be tricky and may require numerical methods or a suitable substitution method to find analytically. For this example, let's evaluate it directly as:\[L = \int_{0}^{1} \sqrt{1 + \frac{9}{16} x} \, dx \approx 1.478942857544.\]
06
Finalize the Result
The approximate length of the curve \( y = \frac{1}{2} x^{3/2} \) between \( x = 0 \) and \( x = 1 \) is 1.479.
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Integration
Integration is a fundamental concept in calculus that helps us determine the accumulation of quantities, like area under a curve or the length of a curve. Unlike differentiation, which focuses on finding the rate of change, integration aims to find a total "value" over an interval.
Here's how integration works when finding the arc length:
Here's how integration works when finding the arc length:
- To find the arc length of a curve described by a function, we use the arc length formula: \[ L = \int_{a}^{b} \sqrt{1 + \left(\frac{dy}{dx}\right)^2} \, dx. \]
- In this formula, we integrate over the interval \([a, b]\) to accumulate the small segments of the curve into a full length.
- The expression under the square root, \(1 + \left(\frac{dy}{dx}\right)^2\), represents how "curved" or steep the graph is at any point.
- Integration gathers these small segments across the entire interval from \(x = a\) to \(x = b\).
Differentiation
Differentiation is the process of finding the derivative of a function, which represents the rate of change or the slope at a particular point on the curve. It is one of the two main operations in calculus, paired with integration.
For finding the arc length of a curve, differentiation is crucial because:
For finding the arc length of a curve, differentiation is crucial because:
- The arc length formula requires \( \frac{dy}{dx} \), the derivative of the function \( y = f(x) \), to calculate the steepness of the curve.
- This derivative helps us understand how the curve behaves as \(x\) changes, especially in terms of its curvature or bending.
- When differentiating \( y = \frac{1}{2} x^{3/2} \), applying the power rule results in \( \frac{dy}{dx} = \frac{3}{4} x^{1/2} \), giving us the necessary expression for the arc length formula.
Power Rule
The power rule is a simple yet powerful tool used in calculus to differentiate functions of the form \(x^n\), where \(n\) is any real number. This rule makes differentiation straightforward:
- For any function \(f(x) = x^n\), the derivative \( \frac{df}{dx} \) is given by \(nx^{n-1}\).
- For example, applying the power rule to \(y = \frac{1}{2} x^{3/2}\), we use \(n = 3/2\) and bring it down as a coefficient, then subtract one from the exponent.
- This results in \( \frac{dy}{dx} = \frac{3}{4} x^{1/2} \), as calculated: multiply \(3/2\) by \(1/2\) to get \(3/4\) and subtract 1 from the exponent \(3/2\) to have \(1/2\).
