Ch. 5 - Integration

Briggs - Calculus: Early Transcendentals 3rd Edition

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Briggs 3rd Edition

Ch. 5 - Integration

Problema 5.4.29

Capitolo 5, Problema 5.4.29

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) = cos 𝓍 on [―π/2 , π/2]

Guida verificata passo dopo passo

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

\frac{1}{(b - a)} \int_{a}^{b} ƒ (𝓍) d 𝓍

Step 2: Substitute the given interval [―π/2, π/2] into the formula. Here, a = ―π/2 and b = π/2. The formula becomes:

\frac{1}{(π / 2 - - π / 2)} \int_{-}^{π} \cos (𝓍) d 𝓍

Step 3: Simplify the denominator of the fraction. The length of the interval [―π/2, π/2] is π. The formula now becomes:

\frac{1}{π} \int_{-}^{π} \cos (𝓍) d 𝓍

Step 4: Compute the integral of cos(𝓍) over the interval [―π/2, π/2]. Recall that the integral of cos(𝓍) is sin(𝓍). Apply the Fundamental Theorem of Calculus:

\frac{1}{π} [\sin (𝓍)] | - π / 2^π / 2

Step 5: Evaluate the definite integral by substituting the limits of integration (―π/2 and π/2) into sin(𝓍). Simplify the result to find the average value of the function. Finally, draw the graph of ƒ(𝓍) = cos(𝓍) on the interval [―π/2, π/2] and indicate the average value as a horizontal line on the graph.

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Average Value of a Function

The average value of a function over a given interval is calculated using the formula (1/(b-a)) * ∫[a to b] f(x) dx, where [a, b] is the interval. This concept helps in understanding how the function behaves on average across the specified range, providing insight into its overall trend rather than just its individual values.

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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. It is denoted as ∫[a to b] f(x) dx and is essential for calculating the total value of a function between two points, which is a key step in finding the average value.

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Definition of the Definite Integral

Graphing Functions

Graphing a function involves plotting its values on a coordinate system, which visually represents its behavior over an interval. This is crucial for understanding the function's characteristics, such as peaks, troughs, and the average value, allowing for a more intuitive grasp of the function's overall performance.

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Graph of Sine and Cosine Function

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Definite integrals Use geometry (not Riemann sums) to evaluate the following definite integrals. Sketch a graph of the integrand, show the region in question, and interpret your result.

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Identifying definite integrals as limits of sums Consider the following limits of Riemann sums for a function ƒ on [a,b]. Identify ƒ and express the limit as a definite integral.

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∫₀ᵃ ƒ(𝓍) d𝓍

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Explain the statement that a continuous function on an interval [a,b] equals its average value at some point on (a,b).

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