Problem 2 - Calculus Early Transcendentals

Question

Explain the steps required to find the length of a curve \(x=g(y)\) between \(y=c\) and \(y=d.\)

Step-by-Step Solution

Answer

Question: Find the length of the curve defined by the equation \(x = g(y)\) between the y values, \(y = c\) and \(y = d\). Answer: The length of the curve, \(L\), is given by \(L = \int_{c}^{d} \sqrt{1+(\frac{dx}{dy})^2} dy\), where \(\frac{dx}{dy}\) is the derivative of \(x = g(y)\) with respect to y. To find the length, follow the steps outlined in the solution above.

Step 1: Find the derivative of x with respect to y

Differentiate the given equation \(x = g(y)\) with respect to y, such that: \(\frac{dx}{dy} = \frac{d}{dy} g(y)\)

Step 2: Square the result and add 1 to it

After finding the derivative, square it and add 1 to the resulting expression: \(1+(\frac{dx}{dy})^2 = 1+ (\frac{d}{dy} g(y))^2\)

Step 3: Take the square root of the expression

Take the square root of the expression obtained in Step 2: \(\sqrt{1+(\frac{dx}{dy})^2} = \sqrt{1+ (\frac{d}{dy} g(y))^2}\)

Step 4: Integrate the expression with respect to y from c to d

Integrate the expression from Step 3 with respect to y, from \(y = c\) to \(y = d\): \(L = \int_{c}^{d} \sqrt{1+(\frac{dx}{dy})^2} dy = \int_{c}^{d} \sqrt{1+(\frac{d}{dy} g(y))^2} dy\)

Step 5: Calculate the Length (L)

Evaluate the integral from Step 4 to find the length of the curve \(x=g(y)\) between \(y=c\) and \(y=d\).

Key Concepts

Integration

Integration is a mathematical process that is essentially the opposite of differentiation. It is used to accumulate quantities, such as finding the area under a curve. In our context, we're using integration to find the length of a curve.
When calculating the arc length, we're essentially summing up an infinite number of very small line segments along the curve. This is done by taking the continuous sum of the curve's infinitesimal lengths. These tiny lengths are represented as \[\sqrt{1+ (\frac{dx}{dy})^2}\] which is derived from the Pythagorean theorem, adapted for a continuous curve.
The integration is set up over the range from \(y=c\) to \(y=d\), where \(y\) is the parameter defining the interval over which we want to find the arc length. This results in the integral:\[L = \int_{c}^{d} \sqrt{1+ (\frac{d}{dy} g(y))^2} \, dy\]This process makes sure we correctly account for the total length by integrating the individual small segments of the curve from start to end.

Differentiation

Differentiation involves finding the derivative of a function, which describes how the function changes as its input changes. In our problem concerning arc length, we focus on differentiation to obtain \(\frac{dx}{dy}\), the rate of change of \(x\) with respect to \(y\).
Given the function \(x = g(y)\), differentiation is used to express this relationship through the equation:\[\frac{dx}{dy} = \frac{d}{dy} g(y)\]This derivative is a crucial part of determining the arc length because it describes the slope of the curve at any point.

• It allows us to account for how steep the curve is.
• Squaring this slope in the formula \(1 + (\frac{dx}{dy})^2\) helps measure the rate at which the curve changes direction.

This understanding of the curve's steepness and change in direction is critical for calculating the arc length accurately.

Curve Analysis

Curve Analysis involves studying the properties and behavior of graphs, particularly focusing on their shapes and slopes. In the context of finding arc length, curve analysis helps us connect the concepts of geometry and calculus.
We analyze the curve given as \(x=g(y)\) to understand how it stretches between two points, \(y=c\) and \(y=d\).

• This analysis helps us visualize the curve.
• Allows us to apply the derivative to determine how the curve behaves.

By examining how the curve behaves between these points, we can effectively use calculus techniques such as differentiation and integration to measure its length.
Ultimately, curve analysis bridges the gap between an abstract mathematical equation and a tangible geometric shape, enabling us to accurately measure the path length along the curve.