CramX Logo
RegisterLog in
Back to FlashcardsMathematics / AP Calculus AB: 9.6.2 An Example of the Trapezoidal Rule

AP Calculus AB: 9.6.2 An Example of the Trapezoidal Rule

Mathematics14 CardsCreated 8 months ago

This section demonstrates how to apply the Trapezoidal Rule to approximate definite integrals, such as estimating ln f(x)=1/x over [1, 3]. It walks through partitioning the interval, calculating function values, and plugging them into the formula, emphasizing how increasing partitions improves accuracy—even when only discrete data points are available.

Report

An Example of the Trapezoidal Rule

• The trapezoidal rule approximates the area A of the region bound by the curve of a continuous function f (x) and the x-axis using N partitions on [a, b].

Rate to track your progress ✦

Tap or swipe ↕ to flip
Swipe ←→Navigate
1/14

Key Terms

Term
Definition

An Example of the Trapezoidal Rule

• The trapezoidal rule approximates the area A of the region bound by the curve of a continuous function f (x) and the x-axis using N partitions on...

note

  • Here’s a strange situation. Suppose you are stranded on a desert island with a group of people, and for some reason you need to know the va...

Use the trapezoidal rule with 4 trapezoids to find an approximation for ∫2 0 1/1+x^2dx.

A≈287/260

Use the trapezoidal rule with 4 trapezoids of equal base to approximate ∫ 2 1 1/x^2dx.

A≈1/8[1+2(4/5)^2+2(2/3)^2+2(4/7)^`2+1/4]

Use the trapezoidal rule with 4 trapezoids to find an expression that approximates ∫4 2 e^x dx.

1/4(e^2+2e^5/2+2e^3+2e^7/2+e^4)

Use the trapezoidal rule with 3 trapezoids to find an approximation for ∫ 2 1 √x^2−1 dx.

A≈8+3√3+2√7/18

Related Flashcard Decks

Mathematics

Apex 2.2.3 Quiz: Graphing Functions

This deck covers key concepts from graphing functions, focusing on understanding graphical representations and their corresponding equations.

9 cards
View Deck
Mathematics

8 Basic Trigonometric Identities

This deck covers the fundamental trigonometric identities, including reciprocal, ratio, and Pythagorean identities.

8 cards
View Deck
Mathematics

AP Calculus AB: 9.1.1 Antidifferentiation

This content provides an overview of antidifferentiation—the process of finding a function whose derivative is a given function. It introduces the concept of antiderivatives, outlines how to reverse common derivative rules, and includes simple examples to illustrate the idea.

8 cards
View Deck
Mathematics

AP Calculus AB: 9.1.2 Antiderivatives of Powers

This content explains how to find antiderivatives of functions involving powers of 𝑥, emphasizing the concept of indefinite integrals and the constant of integration 𝐶. It highlights the idea that functions can have infinitely many antiderivatives and includes example problems demonstrating how to evaluate integrals involving powers and rational expressions.

9 cards
View Deck
Mathematics

AP Calculus AB: 9.2.2 Integrating Polynomials by Substitution

This content explains how to integrate polynomial expressions using integration by substitution, especially when dealing with composite functions. It outlines the step-by-step substitution process, emphasizes correct variable handling, and highlights the importance of rewriting the final answer in terms of the original variable along with including the constant of integration.

12 cards
View Deck
Mathematics

AP Calculus AB: 9.3.4 Choosing Effective Function Decompositions

This section focuses on how to strategically choose a substitution when performing integration by substitution. It explains that the best choice for 𝑢 is typically a part of the integrand whose derivative is also present, making the integral easier to evaluate. It also covers how to simplify trigonometric expressions using identities and emphasizes that effective decomposition is key to simplifying complex integrals.

6 cards
View Deck
Mathematics

AP Calculus AB: 9.3.1 Integrating Composite Trigonometric Functions by Substitution

This content covers the technique of integration by substitution applied to composite trigonometric functions. It explains how to identify the inner function for substitution, handle constant multiples, and ensure complete substitution. Worked examples demonstrate how substitution simplifies complex integrals, with emphasis on converting back to the original variable and verifying results through differentiation.

12 cards
View Deck
Mathematics

Conversion Factors and Problem Solving

This deck covers essential concepts related to rounding rules, significant figures, metric conversions, and basic geometry calculations.

18 cards
View Deck
Mathematics

Unit 1 Progress Check: MCQ Part A

This deck covers 18 questions and answers from a unit progress check, focusing on calculus concepts such as limits, derivatives, and function behavior.

18 cards
View Deck
AP Calculus AB: 9.6.2 An Example of the Trapezoidal Rule
TermDefinition

An Example of the Trapezoidal Rule

• The trapezoidal rule approximates the area A of the region bound by the curve of a continuous function f (x) and the x-axis using N partitions on [a, b].

note

  • Here’s a strange situation. Suppose you are stranded on a desert island with a group of people, and for some reason you need to know the value of the natural log of three. Hey, weirder things have happened!

  • Well, if you remember from Calculus I that the integral of 1/x is equal to the natural log function, you can construct an integral whose area is equal to exactly ln 3.

  • How could you use this information to determine the value of ln 3?

  • Since the only tools you have to compute numbers are what you know, then you will have to approximate the value of ln 3.

  • Here is the trapezoidal rule. If you use this rule, your
    approximation will be better than those of the other people stranded on the island.

  • Here, the interval [a, b] is divided into 4 regions.
    So f (x) = 1/x, N = 4, a = 1, and b = 3.

  • If you wanted an even more precise approximation, you could increase the number of rectangles used in the approximation.

  • Each partition has a base of 1/2. Start by finding what the x-values are at the partitions. Then find their corresponding f (x)-values. Plug these values into the trapezoidal rule.

  • Find a common denominator and add the fractions together.

  • Notice that your answer is very close to the actual value of ln 3.

  • Try doing this example with eight partitions to see if the approximation gets better. Also, compare the trapezoidal rule to the rectangular approximation.

  • Here is another example. This time, all that is given is a table of values. The function itself isn’t known, but the integral can still be approximated using the Trapezoidal Rule.

  • There are eight partitions and the endpoints are a = 2 and b = 5.2. The approximate value of the integral is 20.94, and this was obtained by using only the points in the table that lie on the curve.

Use the trapezoidal rule with 4 trapezoids to find an approximation for ∫2 0 1/1+x^2dx.

A≈287/260

Use the trapezoidal rule with 4 trapezoids of equal base to approximate ∫ 2 1 1/x^2dx.

A≈1/8[1+2(4/5)^2+2(2/3)^2+2(4/7)^`2+1/4]

Use the trapezoidal rule with 4 trapezoids to find an expression that approximates ∫4 2 e^x dx.

1/4(e^2+2e^5/2+2e^3+2e^7/2+e^4)

Use the trapezoidal rule with 3 trapezoids to find an approximation for ∫ 2 1 √x^2−1 dx.

A≈8+3√3+2√7/18

Use the Trapezoidal Rule and the following data to estimate the value of the integral ∫ 2.2 1 y dx.

8.36

Use the Trapezoidal Rule and the following data to estimate the value of the integral ∫ 2.8 0 y dx.

18.88

Use the trapezoidal rule with 3 trapezoids to find an approximation for ∫ 3 2 1/1−x dx.

A≈−7/10

Use the trapezoidal rule with 4 trapezoids to find an approximation for ∫ 2 0 x√x^2+4 dx.

A≈1/8[15+8√2+4√5+√17]

Use the trapezoidal rule with 4 trapezoids to find an approximation for ∫π 0 sinx dx.

A≈π(1+√2)/4

Use the trapezoidal rule with four trapezoids to find an expression that approximates ∫ 3 1 √x dx.

1/4[1+2√3/2+2√2+2√5/2+√3]

Use the trapezoidal rule with 4 trapezoids to find an approximation for ∫1 0 x^3/2dx.

A≈5+2√2+3√3/32

Use the trapezoidal rule with 3 trapezoids to find an approximation for ∫ 1 0 1−x/1+x dx.

A≈2/5