Table of contents

0. Functions7h 52m
1. Limits and Continuity2h 2m
2. Intro to Derivatives1h 33m
3. Techniques of Differentiation3h 18m
4. Applications of Derivatives2h 38m
5. Graphical Applications of Derivatives6h 2m
6. Derivatives of Inverse, Exponential, & Logarithmic Functions2h 37m
7. Antiderivatives & Indefinite Integrals1h 26m
8. Definite Integrals4h 44m
9. Graphical Applications of Integrals2h 27m
- Area Between Curves
  1h 18m
- Introduction to Volume & Disk Method
  1h 8m
10. Physics Applications of Integrals 3h 16m
- Kinematics
  1h 13m
- Work
  2h 3m
11. Integrals of Inverse, Exponential, & Logarithmic Functions2h 34m
12. Techniques of Integration7h 39m
13. Intro to Differential Equations2h 55m
14. Sequences & Series5h 36m
15. Power Series2h 19m
- Introduction to Power Series
  1h 9m
- Taylor Series & Taylor Polynomials
  1h 10m
16. Parametric Equations & Polar Coordinates7h 58m

8. Definite Integrals

Average Value of a Function

Calculus

8. Definite Integrals

Average Value of a Function

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3:00 minutes

Problem 5.4.49

Textbook Question

Symmetry of composite functions Prove that the integrand is either even or odd. Then give the value of the integral or show how it can be simplified. Assume f and g are even functions and p and q are odd functions.

∫ᵃ₋ₐ ƒ(g(𝓍)) d𝓍

Verified step by step guidance

Step 1: Recall the definitions of even and odd functions. An even function satisfies f(x) = f(-x), while an odd function satisfies p(x) = -p(-x). These properties will be crucial in analyzing the symmetry of the composite function f(g(x)).

Step 2: Analyze the composition f(g(x)). Since f(x) is even, f(g(x)) will inherit the evenness of g(x) if g(x) is also even. Specifically, g(x) = g(-x) implies f(g(x)) = f(g(-x)), making f(g(x)) an even function.

Step 3: Consider the integral ∫ᵃ₋ₐ f(g(x)) dx. If f(g(x)) is even, the integral over the symmetric interval [-a, a] simplifies to 2 * ∫₀ᵃ f(g(x)) dx because the contributions from [-a, 0] and [0, a] are identical.

Step 4: If g(x) were odd, g(-x) = -g(x). Substituting this into f(g(x)), we would need to check whether f(-g(x)) = -f(g(x)) (odd) or f(-g(x)) = f(g(x)) (even). This determines whether f(g(x)) is odd or even.

Step 5: Conclude the symmetry analysis. If f(g(x)) is odd, the integral ∫ᵃ₋ₐ f(g(x)) dx evaluates to 0 because the contributions from [-a, 0] and [0, a] cancel each other out. If f(g(x)) is even, simplify the integral as described in Step 3.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Even and Odd Functions

Even functions are symmetric about the y-axis, meaning f(-x) = f(x) for all x. Odd functions have rotational symmetry about the origin, satisfying f(-x) = -f(x). Understanding these properties is crucial for analyzing the symmetry of composite functions and determining the nature of the integrand.

Composite Functions

A composite function is formed when one function is applied to the result of another function, denoted as (f ∘ g)(x) = f(g(x)). In the context of the integral, knowing how to evaluate the symmetry of composite functions helps in determining whether the integrand is even or odd, which influences the integral's value.

Properties of Integrals

The integral of an even function over a symmetric interval [-a, a] is twice the integral from 0 to a, while the integral of an odd function over the same interval is zero. These properties simplify the evaluation of integrals and are essential for proving the nature of the integrand in the given problem.

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Textbook Question

Planetary orbits The planets orbit the Sun in elliptical orbits with the Sun at one focus (see Section 12.4 for more on ellipses). The equation of an ellipse whose dimensions are 2a in the 𝓍-direction and 2b in the y-direction is (𝓍²/a²) + (y² /b²) = 1.

(a) Let d² denote the square of the distance from a planet to the center of the ellipse at (0, 0). Integrate over the interval [ ―a, a] to show that the average value of d² is (a² + 2b²) /3 .

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Textbook Question

Average value with a parameter Consider the function ƒ(𝓍) = a𝓍 (1―𝓍) on the interval [0, 1], where a is a positive real number.

(a) Find the average value of ƒ as a function of a .

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Textbook Question

Suppose ƒ is an odd function, ∫₀⁴ ƒ(𝓍) d𝓍 = 3 , and ∫₀⁸ ƒ(𝓍) d𝓍 = 9 .

(a) Evaluate ∫₋₈⁴ ƒ(𝓍) d𝓍 .

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Textbook Question

Suppose ƒ is an even function and ∫⁸₋₈ ƒ(𝓍) d𝓍 = 18

(b) Evaluate ∫₋₈⁸ 𝓍ƒ(𝓍) d𝓍 .

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Textbook Question

If ƒ is an even function, why is ∫ᵃ₋ₐ ƒ(𝓍) d𝓍 = 2 ∫₀ᵃ ƒ(𝓍) d𝓍?

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Textbook Question

If ƒ is an odd function, why is ∫ᵃ₋ₐ ƒ(𝓍) d𝓍 = 0?

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Textbook Question

Symmetry properties Suppose ∫₀⁴ ƒ(𝓍) d𝓍 = 10 and ∫₀⁴ g(𝓍) d𝓍 = 20. Furthermore, suppose ƒ is an even function and g is an odd function. Evaluate the following integrals.

(e) ∫₋₂² 3𝓍ƒ(𝓍)d𝓍

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Textbook Question

Symmetry properties Suppose ∫₀⁴ ƒ(𝓍) d𝓍 = 10 and ∫₀⁴ g(𝓍) d𝓍 = 20. Furthermore, suppose ƒ is an even function and g is an odd function. Evaluate the following integrals.

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Symmetry of composite functions Prove that the integrand is either even or odd. Then give the value of the integral or show how it can be simplified. Assume f and g are even functions and p and q are odd functions.∫ᵃ₋ₐ ƒ(g(𝓍)) d𝓍

Key Concepts

Even and Odd Functions

Composite Functions

Properties of Integrals

Watch next

Symmetry of composite functions Prove that the integrand is either even or odd. Then give the value of the integral or show how it can be simplified. Assume f and g are even functions and p and q are odd functions.

∫ᵃ₋ₐ ƒ(g(𝓍)) d𝓍