Numerade

Calculus: Early Transcendentals

James Stewart

ISBN #9781285741550

8th Edition

6,422 Questions

2,819,387 Students Helped

Summary

This section introduces the fundamental concept of the definite integral as the limit of sums of areas of approximating rectangles (Riemann sums). Starting from the challenge of defining the area under a curve, the text shows how subdividing a region and refining estimates yields the exact area as a limit. The same ideas apply to estimating distance traveled from variable velocity data. The use of left, right, and midpoint approximations, along with sigma notation, illustrate how calculus transforms an intuitive idea into a precise mathematical tool with wide applications.

Learning Objectives

Explain how the area under a curve can be approximated using Riemann sums and how this leads to the concept of the definite integral.

Demonstrate the construction of lower sums, upper sums, and midpoint approximations, and understand their roles in estimating areas and distances.

Apply the limit of Riemann sums to define the exact area under a continuous function and to compute distances traveled from variable velocity data.

Utilize sigma notation to represent and simplify the sum of areas of rectangles in both theoretical and practical problems.

Key Concepts

CONCEPT

DEFINITION

Riemann Sum

An approximation of the area under a curve using a finite sum of areas of rectangles; each rectangle’s area is given by f(x_i)*Δx, where Δx is the width of subintervals.

Definite Integral

The limit of the Riemann sums as the number of subintervals approaches infinity. It gives the exact area under a curve (or distance traveled) and is denoted by ∫_a^b f(x)dx.

Subinterval

A division of the interval [a, b] into smaller intervals of equal width Δx = (b − a)/n used to construct approximating rectangles.

Left, Right, and Midpoint Endpoints

Sample points chosen respectively at the start, end, or middle of each subinterval to determine the height of the approximating rectangle in a Riemann sum.

Sigma Notation

A compact way of writing sums; for example, the sum of areas f(x_i)*Δx from i = 1 to n is expressed as Σ[i=1 to n] f(x_i)*Δx.

Limit

A fundamental concept in calculus indicating the value that a sequence or function approaches as the index or input approaches a particular value (often infinity).

Example 1

(a) By reading values from the given graph of $ f $, use five rectangles to find a lower estimate and an upper estimate for the area under the given graph of $ f $ from $ x = 0 $ to $ x = 10 $. In each case sketch the rectangles that you use. (b) Find new estimates using ten rectangles in each case.

Example 2

(a) Use six rectangles to find estimates of each type for the area under the given graph of $ f $ from $ x = 0 $ to $ x = 12 $. (i) $ L_{6} $ (sample points are left endpoints) (ii) $ R_{6} $ (sample points are right endpoints) (iii) $ M_{6} $ (sample points are midpoints) (b) Is $ L_{6} $ an underestimate or overestimate of the true area? (c) Is $ R_{6} $ an underestimate or overestimate of the true area? (d) Which of the numbers $ L_{6} $, $ R_{6} $, or $ M_{6} $ gives the best estimate? Explain.

Example 5

(a) Estimate the area under the graph of $ f(x) = 1 + x^2 $ from $ x = -1 $ to $ x = 2 $ using three rectangles and right endpoints. Then improve your estimate by using six rectangles. Sketch the curve and the approximating rectangles. (b) Repeat part (a) using left endpoints. (c) Repeat part (a) using midpoints. (d) From your sketches in parts (a)-(c), which appears to be the best estimate?

Step-by-Step Explanations

QUESTION

How do you approximate the area under the curve f(x)=x² on the interval [0,1] using n rectangles with right endpoints?

STEP-BY-STEP ANSWER:

Step 1: Divide the interval [0, 1] into n equal subintervals, so that each Δx = (1 - 0)/n = 1/n.
Step 2: Identify the right endpoint of the ith subinterval, which is x_i = i/n.
Step 3: Compute the height of each rectangle using the function value at the right endpoint: f(x_i)= (i/n)².
Step 4: Determine the area of the ith rectangle: Area_i = (i/n)² * (1/n).
Step 5: Write the Riemann sum as the total approximate area: R_n = Σ[i=1 to n] (i/n)² * (1/n).
Step 6: Express the exact area as the limit of these sums as n → ∞: Area = lim (n→∞) Σ[i=1 to n] (i/n)² * (1/n).
Final Answer: The definite integral, representing the area, is given by ∫₀¹ x² dx = lim (n→∞) Σ[i=1 to n] (i/n)² * (1/n).

Approximating the Area Under f(x)=x² on [0,1] Using Right Endpoints

QUESTION

How is the distance traveled by an object with velocity function v(t) estimated using Riemann sums?

STEP-BY-STEP ANSWER:

Step 1: Given a velocity function v(t) and a time interval [a, b], divide the interval into n subintervals where Δt = (b − a)/n.
Step 2: Choose sample points in each subinterval (using left endpoints, right endpoints, or midpoints) to approximate the velocity.
Step 3: For each subinterval, estimate the distance traveled as the product of the velocity at the sample point and the time interval: v(t_i*) * Δt.
Step 4: Sum these distances to form an estimate of the total distance traveled: D_n = Σ[i=1 to n] v(t_i*) * Δt.
Step 5: Define the exact distance as the limit of these sums as n becomes large: Distance = lim (n→∞) Σ[i=1 to n] v(t_i*) * Δt.
Final Answer: The total distance traveled is the definite integral of the velocity function, ∫ₐᵇ v(t) dt.

Estimating Distance Traveled with Variable Velocity

Common Mistakes

Failing to correctly compute the width of subintervals as ?x = (b ? a)/n, leading to inaccuracies.

Mixing up endpoint methods; for an increasing function, using left endpoints results in underestimates while right endpoints lead to overestimates.

Incorrectly setting up the sigma notation sum, or omitting the limit process as n ? ?.

Misinterpreting the graphical representation of approximating rectangles, which can lead to errors in visual estimation.