Ch. 5 - Integration

Briggs - Calculus: Early Transcendentals 3rd Edition

Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243Not the one you use?Change textbook

Briggs 3rd Edition

Ch. 5 - Integration

Problem 5.RE.15c

Chapter 5, Problem 5.RE.15c

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.

(c) ∫₋₄⁴ (4ƒ(𝓍) ― 3g(𝓍))d𝓍

Verified step by step guidance

Step 1: Recall the symmetry properties of even and odd functions. An even function satisfies ƒ(𝓍) = ƒ(−𝓍), and its integral over a symmetric interval [−a, a] is twice the integral over [0, a]. An odd function satisfies g(𝓍) = −g(−𝓍), and its integral over a symmetric interval [−a, a] is 0.

Step 2: Break the given integral ∫₋₄⁴ (4ƒ(𝓍) ― 3g(𝓍)) d𝓍 into two separate integrals: ∫₋₄⁴ 4ƒ(𝓍) d𝓍 and ∫₋₄⁴ −3g(𝓍) d𝓍. This uses the linearity property of integrals.

Step 3: For the first term, ∫₋₄⁴ 4ƒ(𝓍) d𝓍, note that ƒ(𝓍) is an even function. Therefore, ∫₋₄⁴ ƒ(𝓍) d𝓍 = 2∫₀⁴ ƒ(𝓍) d𝓍. Multiply this result by 4 to account for the coefficient.

Step 4: For the second term, ∫₋₄⁴ −3g(𝓍) d𝓍, note that g(𝓍) is an odd function. The integral of an odd function over a symmetric interval [−a, a] is 0. Therefore, this term evaluates to 0.

Step 5: Combine the results from Step 3 and Step 4. The final integral ∫₋₄⁴ (4ƒ(𝓍) ― 3g(𝓍)) d𝓍 simplifies to the result obtained from the first term, as the second term is 0.

Verified video answer for a similar problem:

This video solution was recommended by our tutors as helpful for the problem above.

Video duration:

Was this helpful?

Key Concepts

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

Even and Odd Functions

An even function satisfies the property f(-x) = f(x) for all x in its domain, meaning its graph is symmetric about the y-axis. Conversely, an odd function satisfies g(-x) = -g(x), indicating symmetry about the origin. These properties are crucial for evaluating integrals over symmetric intervals, as they allow simplifications based on the behavior of the functions.

Properties of Definite Integrals

Definite integrals have specific properties that can simplify calculations. For instance, 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 help in evaluating integrals without direct computation.

Linear Combination of Integrals

The linearity of integrals allows us to combine integrals of functions through addition and scalar multiplication. Specifically, ∫(af(x) + bg(x))dx = a∫f(x)dx + b∫g(x)dx, where a and b are constants. This property is essential for evaluating integrals involving multiple functions, as it enables the separation of terms for easier computation.

Recommended video:

07:17

Linearization

Related Practice

Textbook Question

Area by geometry Use geometry to evaluate the following definite integrals, where the graph of ƒ is given in the figure.

(b) ∫₆⁴ ƒ(𝓍) d𝓍

views

Textbook Question

Find the intervals on which ƒ(𝓍) = ∫ₓ¹ (t―3) (t―6)¹¹ dt is increasing and the intervals on which it is decreasing.

views

Textbook Question

Area by geometry Use geometry to evaluate the following definite integrals, where the graph of ƒ is given in the figure.

(a) ∫₀⁴ ƒ(𝓍) d𝓍

views

Textbook Question

The velocity in ft/s of an object moving along a line is given by v = ƒ(t) on the interval 0 ≤ t ≤ 6 (see figure), where t is measured in seconds.

(a) Divide the interval [0,6] into n = 3 subintervals, [0,2] , [2,4] and [4,6]. On each subinterval, assume the object moves at a constant velocity equal to the value of v evaluated at the right endpoint of the subinterval, and use these approximations to estimate the displacement of the object on [0,6] (see part (a) of the figure)

views

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

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.

(a) ∫₋₄⁴ ƒ(𝓍) d𝓍

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.(c) ∫₋₄⁴ (4ƒ(𝓍) ― 3g(𝓍))d𝓍

Verified video answer for a similar problem:

Key Concepts

Even and Odd Functions

Properties of Definite Integrals

Linear Combination of Integrals

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.

(c) ∫₋₄⁴ (4ƒ(𝓍) ― 3g(𝓍))d𝓍