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.R.97

Chapter 5, Problem 5.R.97

Find the average value of ƒ(𝓍) = e²ˣ on [0, ln 2] .

Verified step by step guidance

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

. Here, a = 0 and b = ln(2).

Step 2: Substitute the given function ƒ(𝓍) = e²ˣ into the formula. The integral becomes:

\frac{1}{(\ln (2) - 0)} \int e^{2 x} d x

Step 3: Compute the integral of e²ˣ with respect to 𝓍. Use the rule for integrating exponential functions:

\int e^{k x} d x = \frac{e^{k x}}{k}

, where k is a constant. Here, k = 2.

Step 4: Evaluate the definite integral from 𝓍 = 0 to 𝓍 = ln(2). Substitute the limits of integration into the antiderivative obtained in Step 3.

Step 5: Multiply the result of the definite integral by

\frac{1}{\ln}

to find the average value of the function on the interval [0, ln(2)].

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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 f(x) over the interval [a, b] is calculated using the formula (1/(b-a)) * ∫[a to b] f(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 output.

Definite Integral

A definite integral represents the accumulation of quantities, such as area under a curve, over a specific interval [a, b]. It is denoted as ∫[a to b] f(x) dx and is calculated using the Fundamental Theorem of Calculus, which connects differentiation and integration, allowing us to evaluate the integral using antiderivatives.

Exponential Functions

Exponential functions, such as f(x) = e^(kx), where e is Euler's number, are characterized by their constant growth rate proportional to their value. In this case, the function e^(2x) grows rapidly as x increases, and understanding its properties is crucial for evaluating integrals involving exponential terms.

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Exponential Functions

Related Practice

Textbook Question

Area versus net area Find (i) the net area and (ii) the area of the region bounded by the graph of ƒ and the 𝓍-axis on the given interval. You may find it useful to sketch the region.

ƒ(𝓍) = 𝓍⁴ ― 𝓍² on [―1, 1]

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Textbook Question

Function defined by an integral Let H (𝓍) = ∫₀ˣ √(4 ― t²) dt, for ― 2 ≤ 𝓍 ≤ 2.

(a) Evaluate H (0) .

Textbook Question

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

(a) A(𝓍) = ∫ₐˣ ƒ(t) dt and ƒ(t) = 2t―3 , then A is a quadratic function.

Textbook Question

Evaluating integrals Evaluate the following integrals.

∫₀¹ √𝓍 (√𝓍 + 1) d𝓍

Textbook Question

Area of regions Compute the area of the region bounded by the graph of ƒ and the 𝓍-axis on the given interval. You may find it useful to sketch the region.

ƒ(𝓍) = 2 sin 𝓍/4 on [0, 2π]

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Textbook Question

Properties of integrals Suppose ∫₁⁴ ƒ(𝓍) d𝓍 = 6 , ∫₁⁴ g(𝓍) d𝓍 = 4 and ∫₃⁴ ƒ(𝓍) d𝓍 = 2 . Evaluate the following integrals or state that there is not enough information.

∫₁³ ƒ(𝓍)/g(𝓍) d𝓍