Calculus: Early Transcendentals
8th Edition
ISBN:9781285741550
Author:James Stewart
Publisher:James Stewart
Chapter1: Functions And Models
Section: Chapter Questions
Problem 1RCC: (a) What is a function? What are its domain and range? (b) What is the graph of a function? (c) How...
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![Below is the transcribed mathematical expression suitable for an educational website:
\[
2. \int_{1}^{2} \int_{3}^{4} \sqrt{xy} \, dy \, dx = \int_{1}^{2} \sqrt{x} \, dx \int_{3}^{4} \sqrt{y} \, dy
\]
### Explanation:
This expression represents the equality of two iterated integrals. The left side involves integrating the function \(\sqrt{xy}\) first with respect to \(y\) from 3 to 4, and then with respect to \(x\) from 1 to 2. The right side separates the integration into a product of two integrals: one for \(\sqrt{x}\) with respect to \(x\) from 1 to 2, and the other for \(\sqrt{y}\) with respect to \(y\) from 3 to 4.
This equality demonstrates how the integration of a function of two variables can sometimes be separated into simpler integrals when the function can be expressed as a product of single-variable functions.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F76c78a43-228d-485d-a6cf-d30956c84b3b%2F2e5d7e5a-b3be-48c0-ba88-4c7df00a7c29%2F9rejszm_processed.jpeg&w=3840&q=75)
Transcribed Image Text:Below is the transcribed mathematical expression suitable for an educational website:
\[
2. \int_{1}^{2} \int_{3}^{4} \sqrt{xy} \, dy \, dx = \int_{1}^{2} \sqrt{x} \, dx \int_{3}^{4} \sqrt{y} \, dy
\]
### Explanation:
This expression represents the equality of two iterated integrals. The left side involves integrating the function \(\sqrt{xy}\) first with respect to \(y\) from 3 to 4, and then with respect to \(x\) from 1 to 2. The right side separates the integration into a product of two integrals: one for \(\sqrt{x}\) with respect to \(x\) from 1 to 2, and the other for \(\sqrt{y}\) with respect to \(y\) from 3 to 4.
This equality demonstrates how the integration of a function of two variables can sometimes be separated into simpler integrals when the function can be expressed as a product of single-variable functions.
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