Book cover for Calculus: Early Transcendentals

Calculus: Early Transcendentals

James Stewart

ISBN #9781285741550

8th Edition

6,422 Questions

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Summary
Learning Objectives
Key Concepts
Example Problems
Explanations
Common Mistakes

Summary

This textbook section focuses on extending integration methods to compute the arc length of curves and the surface area of surfaces of revolution. By approximating curves with polygonal segments and taking limits, integral formulas for arc length are derived. The section emphasizes both analytical derivations and numerical approximations (like Simpson’s Rule) when explicit evaluation is challenging. The connection between the arc length function and its geometric interpretation, as well as its applications in real-world scenarios, is highlighted throughout.

Learning Objectives

1

Explain the concept of arc length and how it is defined via polygonal approximations and limits.

2

Derive and utilize the integral formulas for computing arc length of curves defined by y = f(x) and x = g(y).

3

Understand the derivation of the arc length function and its properties using the Fundamental Theorem of Calculus.

4

Apply techniques of integration, including substitutions and numerical methods like Simpson’s Rule, to compute arc lengths and surface areas of revolution.

Key Concepts

CONCEPT

DEFINITION

Arc Length

The length of a curve defined as the limit of the lengths of inscribed polygonal approximations as the number of segments approaches infinity.

Polygonal Approximation

A method of approximating a curve by a sequence of connected line segments whose total length converges to the actual arc length as the segments become finer.

Arc Length Formula

For a smooth curve y = f(x) on [a, b] the arc length L is given by L = ∫[a,b] √(1 + [f'(x)]²) dx. A similar formula applies when the function is given in terms of y.

Arc Length Function

A function s(x) that measures the distance along the curve from a fixed starting point to any other point, defined as s(x) = ∫[a,x] √(1 + [f'(t)]²) dt.

Surface of Revolution

A surface formed by rotating a curve about a line (axis). Its lateral surface area is computed using integrals that take into account both the curve's length and the rotation.

Example Problems

Example 1

Use the arc length formula (3) to find the length of the curve $ y = 2x - 5 $, $ -1 \le x \le 3 $. Check your answer by noting that the curve is a line segment and calculating its length by the distance formula.

Example 2

Use the arc length formula to find the length of the curve $ y = \sqrt{2 - x^2} $, $ 0 \le x \le 1 $. Check your answer by noting that the curve is part of a circle.

Example 3

Set up an integral that represents the length of the curve. Then use your calculator to find the length correct to four decimal places. $ y = \sin x $, $ 0 \le x \le \pi $

Example 4

Set up an integral that represents the length of the curve. Then use your calculator to find the length correct to four decimal places. $ y = xe^{-x} $, $ 0 \le x \le 2 $

Example 5

Set up an integral that represents the length of the curve. Then use your calculator to find the length correct to four decimal places. $ y = x - \ln x $, $ 1 \le x \le 4 $
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Step-by-Step Explanations

QUESTION

How can we derive the formula for arc length of a curve y = f(x) from a polygonal approximation?

STEP-BY-STEP ANSWER:

Step 1: Approximate the curve by dividing the interval [a, b] into n subintervals with endpoints x0, x1, ..., xn.
Step 2: On each subinterval, generate the point (xi, f(xi)) and approximate the segment length between consecutive points by using the distance formula.
Step 3: Express the length of each segment as √((Δx)² + (Δy)²), where Δy = f(xi) - f(xi-1).
Step 4: Factor Δx and rewrite the expression as Δx √(1 + (Δy/Δx)²).
Step 5: Recognize that as Δx → 0, Δy/Δx approaches f'(x), and summing the lengths leads to the Riemann sum ∑ Δx √(1 + [f'(x)]²).
Step 6: Take the limit as n → ∞ to convert the Riemann sum into the definite integral: L = ∫[a,b] √(1 + [f'(x)]²) dx.
Final Answer: The arc length is given by L = ∫[a,b] √(1 + [f'(x)]²) dx.

Derivation of the Arc Length Formula for y = f(x)

QUESTION

How is the arc length function used to measure distance along a curve from a fixed starting point?

STEP-BY-STEP ANSWER:

Step 1: Define the arc length function s(x) as the integral from a fixed starting point a to a variable endpoint x.
Step 2: Write s(x) = ∫[a,x] √(1 + [f'(t)]²) dt where t is the dummy variable.
Step 3: Differentiate s(x) with respect to x using the Fundamental Theorem of Calculus to obtain ds/dx = √(1 + [f'(x)]²).
Step 4: Interpret ds/dx as the rate at which the arc length increases with respect to x.
Final Answer: The arc length function s(x) measures the distance along the curve from a to x and its derivative is ds/dx = √(1 + [f'(x)]²).

Using the Arc Length Function

QUESTION

How do we establish an integral to compute the lateral surface area of a surface obtained by rotating a curve about an axis?

STEP-BY-STEP ANSWER:

Step 1: Approximate the curve with small line segments whose lengths are computed via the arc length differential ds.
Step 2: When the curve is rotated about an axis, each segment traces a band shaped like a frustum of a cone.
Step 3: Determine the average radius for the band and multiply by the arc length of the segment.
Step 4: Sum over all segments and take the limit to form the integral for surface area.
Final Answer: The lateral surface area is calculated by an integral of the form A = ∫ (circumference at a point) ds, typically A = ∫[a,b] 2π f(x) √(1 + [f'(x)]²) dx for rotation about the x-axis.

Setting Up an Integral for a Surface of Revolution

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Common Mistakes

  • Assuming that the length of a curved line is simply the straight-line distance between endpoints.
  • Neglecting the proper application of the limit process when approximating a curve by polygons.
  • Forgetting to square the derivative f'(x) when setting up the arc length differential ?(1 + [f'(x)]²).
  • Mixing up the roles of the variables when switching between the functions of x and y in the formulas for arc length.