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 5.4.25

Chapter 5, Problem 5.4.25

Average values Find the average value of the following functions on the given interval. Draw a graph of the function and indicate the average value.
ƒ(𝓍) = 𝓍³ on [―1, 1]

Verified step by step guidance

Step 1: Recall the formula for the average value of a function ƒ(𝓍) on the interval [a, b]. The average value is given by:

\frac{1}{(b - a)} \int f_{x} (x) d x

. Here, a = -1 and b = 1.

Step 2: Substitute the given function ƒ(𝓍) = 𝓍³ into the formula. The integral becomes:

\frac{1}{2} \int (x) x^{3} d x

, where the factor

\frac{1}{2}

comes from

\frac{1}{(}

Step 3: Compute the definite integral of 𝓍³ over the interval [―1, 1]. The integral of 𝓍³ is

\frac{1}{4} x^{4}

. Evaluate this expression at the bounds ―1 and 1.

Step 4: Subtract the value of the integral at the lower bound (―1) from the value at the upper bound (1). This gives:

(\frac{1}{4} 1^{4}) - (\frac{1}{4} -^{14})

Step 5: Multiply the result of the definite integral by

\frac{1}{2}

to find the average value of the function. Finally, draw the graph of ƒ(𝓍) = 𝓍³ and indicate the average value as a horizontal line across the interval [―1, 1].

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Key Concepts

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

Average Value of a Function

The average value of a continuous function over a closed interval [a, b] is calculated using the formula (1/(b-a)) * ∫[a to b] f(x) dx. This represents the mean value of the function across the specified interval, providing insight into the function's overall behavior rather than just its individual points.

Definite Integral

A definite integral computes the accumulation of a quantity, represented as the area under the curve of a function f(x) from a to b. It is denoted as ∫[a to b] f(x) dx and is fundamental in finding the average value, as it quantifies the total output of the function over the interval.

Graphing Functions

Graphing a function involves plotting its output values against input values on a coordinate system, which visually represents the function's behavior. For the function f(x) = x³, the graph will show a cubic curve, and marking the average value on this graph helps illustrate how the average compares to the function's actual values over the interval.

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Related Practice

Textbook Question

Indefinite integrals Use a change of variables or Table 5.6 to evaluate the following indefinite integrals. Check your work by differentiating.

∫ [(√𝓍 + 1)⁴ / 2√𝓍 d𝓍

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Textbook Question

Indefinite integrals Use a change of variables or Table 5.6 to evaluate the following indefinite integrals. Check your work by differentiating.

∫ (𝓍⁶ ― 3𝓍²)⁴ (𝓍⁵ ― 𝓍) d𝓍

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Textbook Question

Average values Find the average value of the following functions on the given interval. Draw a graph of the function and indicate the average value.

ƒ(𝓍) = 𝓍ⁿ on [0,1] , for any positive integer n

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Textbook Question

Area versus net area Graph the following functions. Then use geometry (not Riemann sums) to find the area and the net area of the region described.

The region between the graph of y = 1 - |x| and the x-axis, for -2 ≤ x ≤ 2

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Textbook Question

Definite integrals from graphs The figure shows the areas of regions bounded by the graph of ƒ and the 𝓍-axis. Evaluate the following integrals.

∫ₐ⁰ ƒ(𝓍) d𝓍

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Textbook Question

Identifying Riemann sums Fill in the blanks with an interval and a value of n.

∑ ƒ (1 + k) • 1 is a right Riemann sum for f on the interval [ ___ , ___ ] with

k = 1

n = ________ .

101

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Average values Find the average value of the following functions on the given interval. Draw a graph of the function and indicate the average value.ƒ(𝓍) = 𝓍³ on [―1, 1]

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Key Concepts

Average Value of a Function

Definite Integral

Graphing Functions

Average values Find the average value of the following functions on the given interval. Draw a graph of the function and indicate the average value.
ƒ(𝓍) = 𝓍³ on [―1, 1]