Book cover for Thomas Calculus

Thomas Calculus

George B. Thomas, Jr.

ISBN #9780321878960

13th Edition

6,812 Questions

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Summary
Learning Objectives
Key Concepts
Example Problems
Explanations
Common Mistakes

Summary

This section introduces the fundamental method of integration through finite sum approximations. By partitioning an interval into small subintervals and summing products of function values and subinterval widths, one can approximate areas under curves, distances traveled, and the average value of functions. As the subintervals become thinner (n increases), the finite sums converge to the definite integral, establishing a profound connection between integration and antiderivatives. Additionally, the use of sigma notation provides a compact way of representing these sums, a key practice in calculus.

Learning Objectives

1

Explain how finite sums are used to approximate areas under curves, distances traveled, and the average value of functions.

2

Understand the difference between upper sums, lower sums, and the midpoint rule in numerical integration.

3

Demonstrate how integration is derived as the limit of finite sums with increasingly thinner subintervals.

4

Describe the connection between integration and antiderivatives, and how sigma notation simplifies finite sum expressions.

Key Concepts

CONCEPT

DEFINITION

Finite Sum

A sum that approximates a quantity (such as area) by breaking a region into small pieces and adding the contributions from each piece.

Definite Integral

The limit of finite sum approximations as the number of subintervals approaches infinity, used to calculate areas, volumes, and other accumulative quantities.

Upper Sum

An approximation obtained by taking the maximum value of the function on each subinterval, which typically overestimates the true area under a curve.

Lower Sum

An approximation obtained by taking the minimum value of the function on each subinterval, which typically underestimates the true area.

Midpoint Rule

A method of approximation where the function is evaluated at the midpoint of each subinterval to estimate the area, often yielding a more balanced estimate.

Δx (Delta x)

The width of each subinterval in a partition of an interval [a, b], given by (b - a)/n.

Sigma Notation

A concise notation for expressing sums, using the Greek letter Σ to indicate the addition of a sequence of terms.

Antiderivative

A function whose derivative is the given function; integral calculus establishes a connection between definite integrals and antiderivatives.

Example Problems

Example 1

In Exercises $1-4,$ use finite approximations to estimate the area under the graph of the function using $$\begin{array}{l}{\text { a. a lower sum with two rectangles of equal width. }} \\ {\text { b. a lower sum with four rectangles of equal width. }} \\ {\text { c. an upper sum with two rectangles of equal width. }} \\ {\text { d. an upper sum with four rectangles of equal width. }}\end{array}$$ $f(x)=x^{2}$ between $x=0$ and $x=1$

Example 2

In Exercises $1-4,$ use finite approximations to estimate the area under the graph of the function using $$\begin{array}{l}{\text { a. a lower sum with two rectangles of equal width. }} \\ {\text { b. a lower sum with four rectangles of equal width. }} \\ {\text { c. an upper sum with two rectangles of equal width. }} \\ {\text { d. an upper sum with four rectangles of equal width. }}\end{array}$$ $f(x)=x^{3}$ between $x=0$ and $x=1$

Example 3

In Exercises $1-4,$ use finite approximations to estimate the area under the graph of the function using $$\begin{array}{l}{\text { a. a lower sum with two rectangles of equal width. }} \\ {\text { b. a lower sum with four rectangles of equal width. }} \\ {\text { c. an upper sum with two rectangles of equal width. }} \\ {\text { d. an upper sum with four rectangles of equal width. }}\end{array}$$ $f(x)=1 / x$ between $x=1$ and $x=5$

Example 4

In Exercises $1-4,$ use finite approximations to estimate the area under the graph of the function using $$\begin{array}{l}{\text { a. a lower sum with two rectangles of equal width. }} \\ {\text { b. a lower sum with four rectangles of equal width. }} \\ {\text { c. an upper sum with two rectangles of equal width. }} \\ {\text { d. an upper sum with four rectangles of equal width. }}\end{array}$$ $f(x)=4-x^{2}$ between $x=-2$ and $x=2$

Example 5

Using rectangles each of whose height is given by the value of the function at the midpoint of the rectangle's base (the midpoint rule), estimate the area under the graphs of the following functions, using first two and then four rectangles. $f(x)=x^{2}$ between $x=0$ and $x=1$
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Step-by-Step Explanations

QUESTION

Estimate the area under the curve y = 1 - x² on the interval [0, 1] using two rectangles (upper sum).

STEP-BY-STEP ANSWER:

Step 1: Divide the interval [0, 1] into 2 equal subintervals. Compute Δx = (1 - 0)/2 = 1/2.
Step 2: For an upper sum with a decreasing function, use the left endpoints. Evaluate f(0) = 1 and f(0.5) = 1 - (0.5)² = 0.75.
Step 3: Calculate the area of each rectangle: first rectangle = 1 × (1/2) = 0.5, second rectangle = 0.75 × (1/2) = 0.375.
Step 4: Sum the areas: 0.5 + 0.375 = 0.875.
Final Answer: The estimated area is 0.875 (an upper sum approximation).

Approximating Area Under a Curve

QUESTION

Using a velocity function, estimate the distance traveled over a time interval using finite sums.

STEP-BY-STEP ANSWER:

Step 1: Partition the time interval [a, b] into n equal subintervals with Δt = (b - a)/n.
Step 2: Choose a sample point in each subinterval (e.g., left endpoints) and evaluate the velocity function y(t) at these points.
Step 3: Approximate the distance on each subinterval by multiplying the velocity by Δt (distance ≈ y(tₖ)·Δt).
Step 4: Sum all these approximated distances: D ≈ y(t₁)Δt + y(t₂)Δt + ... + y(tₙ)Δt.
Final Answer: The total estimated distance is the sum computed, which becomes more accurate as n increases.

Estimating Distance Traveled

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Common Mistakes

  • Confusing upper sums with lower sums, leading to incorrect overestimates or underestimates of areas.
  • Assuming that a finite sum approximation exactly equals the true value without considering the error which diminishes as the number of subintervals increases.
  • Mixing up displacement and distance traveled by not using the absolute value of the velocity function when necessary.
  • Incorrectly calculating the width of subintervals (?x or ?t) which can lead to errors in the overall sum.
  • Misapplying sigma notation by not setting the correct limits or index, thus causing misrepresentation of the sums.