CramX Logo
RegisterLog in
Back to FlashcardsMathematics / AP Calculus AB: 10.2.1 The Area between Two Curves

AP Calculus AB: 10.2.1 The Area between Two Curves

Mathematics7 CardsCreated 8 months ago

This section explains how to calculate the area between two curves by integrating the difference between the upper and lower functions over a given interval. It emphasizes correct setup of the integral, using limits of integration from x-values where the region begins and ends, and applying the Fundamental Theorem of Calculus for evaluation.

Report

The Area between Two Curves

  • The definite integral can be used to calculate the area between a curve and the x-axis on a given interval.

  • To find the area of a region bounded by the graphs of two functions, take the definite integral of the difference of the two functions on the appropriate interval.

Rate to track your progress ✦

Tap or swipe ↕ to flip
Swipe ←→Navigate
1/7

Key Terms

Term
Definition

The Area between Two Curves

  • The definite integral can be used to calculate the area between a curve and the x-axis on a given interval.

  • To find the area...

note

  • Notice that the area between these two regions is equal to the area underneath the top curve minus the area underneath the bottom curve. Si...

What is the area between the graphs of the functions f (x) = 5 sin x and g (x) = e ^x − 3, between x = 1 and x = 2?

2∫1 (5sinx−e^x−3)dx

What is the area between the curves f(x)=2−x and g(x)=x^2 between x=−2 and x=1?

A=4 1/2

What is the area between the functions g(x)=cos2x and h(x)=5x, between x=3 and x=4?s

.804

Which of the following integrals represents the area between the graphs of the functions f (x) = |x| and g (x) = x^ 2 + 5, between x = −1 and x = 2?

2∫−1 [x^2+5−|x|]dx

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: 10.2.1 The Area between Two Curves
TermDefinition

The Area between Two Curves

  • The definite integral can be used to calculate the area between a curve and the x-axis on a given interval.

  • To find the area of a region bounded by the graphs of two functions, take the definite integral of the difference of the two functions on the appropriate interval.

note

  • Notice that the area between these two regions is equal to the area underneath the top curve minus the area underneath the bottom curve. Since the sum of two integrals is equal to the integral of the sum, you can combine the two integrals into one.

  • Once you set up the integral, you can use the
    fundamental theorem of calculus to evaluate it.

  • When setting up the integral, remember to take the equation for the upper curve and subtract the equation for the lower curve.

  • The limits of integration are the x-values where the region begins and ends.

  • Notice that the integral itself is very basic. Once simplified, it is just the integral of a polynomial equation. You can evaluate the integral piece by piece.

What is the area between the graphs of the functions f (x) = 5 sin x and g (x) = e ^x − 3, between x = 1 and x = 2?

2∫1 (5sinx−e^x−3)dx

What is the area between the curves f(x)=2−x and g(x)=x^2 between x=−2 and x=1?

A=4 1/2

What is the area between the functions g(x)=cos2x and h(x)=5x, between x=3 and x=4?s

.804

Which of the following integrals represents the area between the graphs of the functions f (x) = |x| and g (x) = x^ 2 + 5, between x = −1 and x = 2?

2∫−1 [x^2+5−|x|]dx

What is the area between the curves f (x) = 3 and g (x) = x ^2 + 4 between x = 0 and x = 3?

A = 12