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Ch. 5 - Integration
Briggs - Calculus: Early Transcendentals 3rd Edition
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Tutti i libri di testoBriggs 3rd EditionCh. 5 - IntegrationProblema 5.4.53a
Capitolo 5, Problema 5.4.53a

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 .

Guida verificata passo dopo passo
1
Step 1: Recall the formula for the average value of a function ƒ(𝓍) on the interval [a, b], which is given by: 1(b-a)∫fxdx. In this case, the interval is [0, 1] and the function is ƒ(𝓍) = a𝓍(1 - 𝓍).
Step 2: Substitute the interval [0, 1] and the function ƒ(𝓍) = a𝓍(1 - 𝓍) into the formula for average value: 11∫0^1(ax(1-x))dx. This simplifies to: ∫0^1ax(1-x)dx.
Step 3: Expand the integrand a𝓍(1 - 𝓍) to simplify the integral. This becomes: a(x-x2). The integral now looks like: a∫0^1(x-x2)dx.
Step 4: Break the integral into two separate parts: a(∫0^1xdx-∫0^1x2dx). Compute each integral separately: ∫0^1xdx and ∫0^1x2dx. Use the power rule for integration: ∫xndx=xn+1n+1.
Step 5: After computing the integrals, combine the results and multiply by the constant 'a' to find the average value of ƒ(𝓍) as a function of 'a'. The final expression will represent the average value of the function over the interval [0, 1].

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Average Value of a Function

The average value of a continuous function ƒ over an interval [a, b] is calculated using the formula (1/(b-a)) * ∫[a to b] ƒ(x) dx. This concept is essential for determining how the function behaves on the specified interval, providing a single representative value that summarizes the function's overall trend.
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Average Value of a Function

Definite Integral

A definite integral represents the accumulation of quantities, such as area under a curve, over a specific interval. In this context, it is used to compute the integral of the function ƒ(x) = a𝓍(1 - 𝓍) from 0 to 1, which is necessary for finding the average value of the function.
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Definition of the Definite Integral

Parameter in Functions

A parameter is a variable that influences the behavior of a function but is not the primary variable of interest. In this case, 'a' is a parameter that affects the shape and scale of the function ƒ(x) = a𝓍(1 - 𝓍), and understanding its role is crucial for expressing the average value as a function of 'a'.
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Eliminating the Parameter
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