Ch. 5 - Integration

Briggs - Calculus: Early Transcendentals 3rd Edition

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Briggs 3rd Edition

Ch. 5 - Integration

문제 5.4.25

5장, 문제 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]

검증된 단계별 안내

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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주요 개념

질문에 올바르게 답하기 위해 반드시 이해해야 하는 핵심 개념들은 다음과 같습니다.

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.

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]