Book cover for Thomas Calculus

Thomas Calculus

George B. Thomas, Jr.

ISBN #9780321878960

13th Edition

6,812 Questions

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127,035 Students Helped

Homework Questions

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Summary
Learning Objectives
Key Concepts
Example Problems
Explanations
Common Mistakes

Summary

This chapter extends the concept of integration from simple coordinate lines and planes to more general curves and surfaces, introducing line integrals as a tool to calculate physical quantities like work and mass along curved paths. It bridges single-variable integration concepts with multivariate applications, and connects the traditional Fundamental Theorem of Calculus to more complex integrals across curves and surfaces. Furthermore, the section introduces vector fields and demonstrates how they are essential in analyzing fluid flow, gravitational and electric forces, and numerous engineering problems.

Learning Objectives

1

Demonstrate how to evaluate line integrals along space curves using parametrizations and the arc length differential.

2

Explain the extension of the Fundamental Theorem of Calculus to curves and surfaces via line and surface integrals.

3

Apply line integrals to compute physical quantities such as work done by a force and mass of wires with variable density.

4

Understand the concept and formulation of vector fields and their role in calculating work and fluid flux.

Key Concepts

CONCEPT

DEFINITION

Line Integral

An integral where a function is evaluated along a curve by summing contributions over small arc segments; defined as the limit of Riemann sums over a partition of the curve.

Parametrization

A representation of a curve in space as r(t) = g(t)i + h(t)j + k(t)k for t in [a, b], which enables evaluation of integrals along that curve.

Arc Length Differential (ds)

An infinitesimal element of length along a curve given by ds = |v(t)| dt, where v(t) is the velocity vector (derivative of r(t)).

Vector Field

A function that assigns a vector to every point in its domain. In three dimensions it is given by F(x, y, z) = M(x, y, z)i + N(x, y, z)j + P(x, y, z)k.

Work Done by a Force

A physical application of line integrals where the work is computed as the line integral of the force vector field along a path.

Example Problems

Example 1

Match the vector equations in Exercises $1-8$ with the graphs (a)-(h) given here. $$ \mathbf{r}(t)=t \mathbf{i}+(1-t) \mathbf{j}, \quad 0 \leq t \leq 1 $$

Example 2

Match the vector equations in Exercises $1-8$ with the graphs (a)-(h) given here. $$ \mathbf{r}(t)=\mathbf{i}+\mathbf{j}+t \mathbf{k}, \quad-1 \leq t \leq 1 $$

Example 3

Match the vector equations in Exercises $1-8$ with the graphs (a)-(h) given here. $$ \mathbf{r}(t)=(2 \cos t) \mathbf{i}+(2 \sin t) \mathbf{j}, \quad 0 \leq t \leq 2 \pi $$

Example 4

Match the vector equations in Exercises $1-8$ with the graphs (a)-(h) given here. $$ \mathbf{r}(t)=t \mathbf{i}, \quad-1 \leq t \leq 1 $$

Example 5

Match the vector equations in Exercises $1-8$ with the graphs (a)-(h) given here. $$ \mathbf{r}(t)=t \mathbf{i}+t \mathbf{j}+t \mathbf{k}, \quad 0 \leq t \leq 2 $$
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Step-by-Step Explanations

QUESTION

How do you compute the line integral of f(x, y, z) = x - 3y^2 + z over the line segment joining the origin to the point (1, 1, 1)?

STEP-BY-STEP ANSWER:

Step 1: Parametrize the line segment by choosing r(t) = t i + t j + t k, where t runs from 0 to 1.
Step 2: Substitute the parametrization into the function, so that f(r(t)) becomes f(t, t, t) = t - 3t^2 + t.
Step 3: Compute the derivative of r(t) to determine ds. Here, r'(t) = i + j + k, and thus |r'(t)| = sqrt(1^2+1^2+1^2) = √3.
Step 4: Write the line integral as ∫[0 to 1] f(r(t)) |r'(t)| dt = √3 ∫[0 to 1] (2t - 3t^2) dt.
Step 5: Evaluate the integral ∫[0 to 1](2t - 3t^2) dt = [t^2 - t^3] evaluated from 0 to 1 = (1 - 1) - (0 - 0) = 0.
Final Answer: The line integral is 0.

Evaluating a Line Integral over a Straight-Line Segment

QUESTION

How do you compute the line integral of f(x, y, z) = 2xy + 2z over the helix r(t)= cos t i + sin t j + t k for t from 0 to π?

STEP-BY-STEP ANSWER:

Step 1: Parametrize the helix. Use r(t) = cos t i + sin t j + t k.
Step 2: Compute r'(t) = -sin t i + cos t j + k, and find its magnitude |r'(t)| = √(sin²t + cos²t + 1) = √2.
Step 3: Substitute the parametrization into the function yielding f(r(t)) = 2 cos t sin t + 2t = sin(2t) + 2t.
Step 4: Write the line integral as ∫[0 to π] [sin(2t) + 2t] √2 dt.
Step 5: Evaluate the definite integral using standard techniques for each term.
Final Answer: The computed value is (√2 multiplied by the evaluated integral), demonstrating an application of the method.

Evaluating a Line Integral over a Helix

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Common Mistakes

  • Misinterpreting the parametrization: Failing to correctly parametrize the curve or using a non-smooth parametrization that causes |r'(t)| to vanish.
  • Incorrect evaluation of the differential ds: Forgetting to compute or include the magnitude of the derivative |r'(t)| can lead to errors.
  • Assuming additivity without proper caution: When a curve is partitioned, neglecting that different paths might yield different integrals if the function is not path-independent.
  • Overlooking the directionality: The parameter t determines the direction along the path, and reversing the direction may change the sign for certain integrals (e.g., in work calculations).