Ch. 5 - Integration

Briggs - Calculus: Early Transcendentals 3rd Edition

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Briggs 3rd Edition

Ch. 5 - Integration

Problem 5.4.53a

Chapter 5, Problem 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 .

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

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

. 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:

\frac{1}{1} \int 0^1 (a x (1 - x)) d x

. This simplifies to:

\int 0^1 a x (1 - x) d x

Step 3: Expand the integrand a𝓍(1 - 𝓍) to simplify the integral. This becomes:

a (x - x^{2})

. The integral now looks like:

a \int 0^1 (x - x^{2}) d x

Step 4: Break the integral into two separate parts:

a (\int 0^1 x d x - \int 0^1 x^{2} d x)

. Compute each integral separately:

\int 0^1 x d x

and

\int 0^1 x^{2} d x

. Use the power rule for integration:

\int x^{n} d x = \frac{x^{n + 1}}{n + 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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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 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.

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.

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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Textbook Question

Bounds on an integral Suppose ƒ is continuous on [a, b] with ƒ''(𝓍) > 0 on the interval. It can be shown that (b―a) ƒ [(a + b) /2] ≤ ∫ₐᵇ ƒ(𝓍) d𝓍 ≤ (b―a) [ (ƒ(a) + ƒ(b)) /2]

(a) Assuming ƒ is nonnegative on [a, b], draw a figure to illustrate the geometric meaning of these inequalities. Discuss your conclusions. b.

Textbook Question

Explain why or why not Determine whether the following statements are true and give an explanation or counterexample. Assume ƒ, ƒ', and ƒ'' are continuous functions for all real numbers.

(a) ∫ ƒ(𝓍) ƒ'(𝓍) d𝓍 = ½ (ƒ(𝓍))² + C.

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Textbook Question

Mass from density A thin 10-cm rod is made of an alloy whose density varies along its length according to the function shown in the figure. Assume density is measured in units of g/cm. In Chapter 6, we show that the mass of the rod is the area under the density curve.

(a) Find the mass of the left half of the rod (0 ≤ x ≤ 5) .

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Textbook Question

Explain why or why not Determine whether the following statements are true and give an explanation or counterexample.

(a) Consider the linear function ƒ(𝓍) = 2x + 5 and the region bounded by its graph and the x-axis on the interval [3,6]. Suppose the area of this region is approximated using midpoint Riemann sums. Then the approximations give the exact area of the region for any number of subintervals.

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Textbook Question

Area functions for constant functions Consider the following functions ƒ and real numbers a (see figure).

(a) Find and graph the area function A(𝓍) = ∫ₐˣ ƒ(t) dt for ƒ.

ƒ(t) = 5 , a = 0

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Textbook Question

Explain why or why not Determine whether the following statements are true and give an explanation or counterexample.

(a) If ƒ is a constant function on the interval [a,b], then the right and left Riemann sums give the exact value of ∫ₐᵇ ƒ(𝓍) d𝓍, for any positive integer n.

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 .

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Key Concepts

Average Value of a Function

Definite Integral

Parameter in Functions

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 .