Numerade

Calculus Early Transcendentals 2

James Stewart

ISBN #9781285741550

8th Edition

6,468 Questions

62,208 Students Helped

Summary

This section introduces the concept of approximating areas under curves using rectangles, leading to the definition of the definite integral as the limit of a Riemann sum. Various methods for choosing sample points (left endpoint, right endpoint, midpoints) are discussed, and it is shown that as the number of rectangles increases, the approximation improves. The approach is not only used to compute areas but also to solve practical problems such as calculating the total distance traveled when velocity is variable.

Learning Objectives

Explain how to approximate the area under a curve by subdividing the region into rectangles and taking limits (Riemann sums).

Define and interpret the definite integral as the limit of Riemann sums, and relate it to areas and distances.

Differentiate between left endpoint, right endpoint, and midpoint approximations and understand their respective roles in overestimation and underestimation.

Apply summation (sigma) notation to express the approximating sums and understand how to transition to the limit form.

Relate the concept of area under a curve to real-world problems such as computing the distance traveled from a velocity function.

Key Concepts

CONCEPT

DEFINITION

Riemann Sum

An approximation of the area under a curve by summing the areas of a finite number of rectangles whose heights are determined by the function's values at specific sample points.

Definite Integral

The limit of the Riemann sums as the number of subintervals approaches infinity. It represents the exact area under a continuous function over a closed interval [a, b] and is denoted by ∫[a,b] f(x) dx.

Subinterval (Δx)

The width of each interval when the region [a, b] is divided into a finite number of equally spaced intervals, calculated as Δx = (b - a)/n.

Sample Point

A point chosen in each subinterval (e.g., left endpoint, right endpoint, or any point within the interval) used to determine the height of the approximating rectangle.

Sigma Notation

A compact notation to express the sum of a sequence of terms, often used to represent Riemann sums in the form ∑ f(x_i*)Δx.

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 $x=12$. $$ \begin{array}{l}{\text { (i) } L_{6} \text { (sample points are left endpoints) }} \\ {\text { (ii) } R_{6} \text { (sample points are right endpoints) }} \\ {\text { (iii) } M_{6} \text { (sample points are midpoints) }}\end{array} $$ (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 esti- mate? 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 we approximate the area under the curve f(x) = x² from x = 0 to x = 1 using n rectangles with right endpoints?

STEP-BY-STEP ANSWER:

Step 1: Divide the interval [0, 1] into n equal subintervals. The width of each subinterval is Δx = 1/n.
Step 2: Identify the right endpoints of each subinterval. These are: x₁ = 1/n, x₂ = 2/n, …, xₙ = n/n = 1.
Step 3: Evaluate the function at each right endpoint to get the height of each rectangle: f(x_i) = (x_i)².
Step 4: Write the sum of the areas of the rectangles as: Rₙ = Σ (from i=1 to n) f(x_i) Δx = Σ (from i=1 to n) (i/n)² (1/n).
Step 5: Express the approximate area as the limit: A = lim (n→∞) Rₙ = lim (n→∞) Σ (i=1 to n) (i²/n³).
Final Answer: The area under the curve is given by A = ∫₀¹ x² dx = lim (n→∞) Σ (i=1 to n) (i²/n³).

Approximating Area Using Right Endpoints

QUESTION

How do we estimate the distance traveled when the velocity is known at discrete times?

STEP-BY-STEP ANSWER:

Step 1: Divide the time interval [a, b] into n subintervals of equal time Δt.
Step 2: Choose a sample point in each time interval (typically the left endpoint, right endpoint, or the midpoint) to approximate the velocity during that interval.
Step 3: Multiply the velocity at the sample point by Δt to estimate the distance covered in that subinterval.
Step 4: Sum the distances for all subintervals: Sₙ = Σ (from i=1 to n) f(t_i*) Δt.
Step 5: Take the limit as n → ∞ to get the exact distance: d = lim (n→∞) Sₙ = ∫[a,b] v(t) dt.
Final Answer: The distance traveled is the definite integral of the velocity function, d = ∫[a,b] v(t) dt.

Estimating Distance Traveled Using Velocity Data

Common Mistakes

Using inconsistent endpoints when subdividing the interval, leading to incorrect approximations.

Forgetting to adjust the width (?x) of each rectangle when the interval [a, b] is subdivided.

Assuming that the approximation is exact without taking the limit as n ? ?.

Mixing up the choice of sample points (left, right, or midpoint) and not recognizing how they affect overestimation or underestimation.

Calculus Early Transcendentals 2

Summary

Example 1

Example 2

Example 3

Example 4

Example 5

Common Mistakes