Question

Determine whether the statement is true or false. If it is true, explain why it is true. If it is false, give an example to show why it is false. If $f$ and $g$ are integrable, then $\int f(x) g(x) d x=$ $\left[\int f(x) d x\right]\left[\int g(x) d x\right]$

Name: Problem 100, Chapter 11: Applied Mathematics: For the Managerial, Life, and Social Sciences
Uploaded: 2021-05-10T14:03:16-08:00
Duration: 1 min 44 s
Channel: Gregory Higby
Description: Determine whether the statement is true or false. If it is true, explain why it is true. If it is false, give an example to show why it is false. If f and g are integrable, then ∫f(x) g(x) d x= [∫f(x) d x][∫g(x) d x]

   Determine whether the statement is true or false. If it is true, explain why it is true. If it is false, give an example to show why it is false.
If $f$ and $g$ are integrable, then $\int f(x) g(x) d x=$ $\left[\int f(x) d x\right]\left[\int g(x) d x\right]$

Applied Mathematics: For the Managerial, Life, and Social Sciences

Soo T. Tan 5th Edition

Chapter 11, Problem 100

Instant Answer

Solved by Expert Gregory Higby

05/10/2021

Step 1

Counterexample: Consider the functions $f(x) = x$ and $g(x) = x$ on the interval $[0, 1]$. Both functions are integrable on this interval. Now, let's compute the integrals: Show more…

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Determine whether the statement is true or false. If it is true, explain why it is true. If it is false, give an example to show why it is false. If $f$ and $g$ are integrable, then $\int f(x) g(x) d x=$ $\left[\int f(x) d x\right]\left[\int g(x) d x\right]$

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Key Concepts

Linearity of Integration

The integration operator is linear, meaning that the integral of a sum of functions equals the sum of their integrals and a constant factor can be pulled out of the integral. However, linearity does not imply that integration distributes over multiplication of functions.

Non-Multiplicative Nature of Integration

In general, the integral of the product of two functions is not equal to the product of their individual integrals. This is a key property of the integration process, as the operation does not preserve the multiplicative structure inherent in the functions.

Role of Counterexamples

Counterexamples illustrate that while both functions may be integrable, the equality between the integral of their product and the product of their integrals fails in general. Constructing simple examples with elementary functions can clearly demonstrate the falsehood of the given statement.

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