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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![### Triple Integral Evaluation
Evaluate the following triple integral:
\[
\int_{-3}^{3} \int_{-\sqrt{9-x^2}}^{\sqrt{9-x^2}} \int_{0}^{\sqrt{9-x^2-y^2}} e^{-(x^2 + y^2 + z^2)^{3/2}} \, dz \, dy \, dx.
\]
This expression represents a triple integral in Cartesian coordinates, where \(x\), \(y\), and \(z\) are integrated over specific bounds. The function \(e^{-(x^2 + y^2 + z^2)^{3/2}}\) is integrated over a three-dimensional region. The limits of integration suggest the region is a semi-spherical volume.
- **\(x\) bounds**: from \(-3\) to \(3\).
- **\(y\) bounds**: for each \(x\), from \(-\sqrt{9-x^2}\) to \(\sqrt{9-x^2}\).
- **\(z\) bounds**: for each pair \((x, y)\), from \(0\) to \(\sqrt{9-x^2-y^2}\).
The integral evaluates the volume under the surface defined by the exponential function within the specified region, which geometrically could represent a hemisphere or similar shape suspended above the xy-plane.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Ff237bbf8-6485-48b1-b1a9-4cbf0285c93d%2Ff2eca5d5-e97c-4b9d-b3b8-aaae97f0ddcb%2Fvize7oi_processed.png&w=3840&q=75)
Transcribed Image Text:### Triple Integral Evaluation
Evaluate the following triple integral:
\[
\int_{-3}^{3} \int_{-\sqrt{9-x^2}}^{\sqrt{9-x^2}} \int_{0}^{\sqrt{9-x^2-y^2}} e^{-(x^2 + y^2 + z^2)^{3/2}} \, dz \, dy \, dx.
\]
This expression represents a triple integral in Cartesian coordinates, where \(x\), \(y\), and \(z\) are integrated over specific bounds. The function \(e^{-(x^2 + y^2 + z^2)^{3/2}}\) is integrated over a three-dimensional region. The limits of integration suggest the region is a semi-spherical volume.
- **\(x\) bounds**: from \(-3\) to \(3\).
- **\(y\) bounds**: for each \(x\), from \(-\sqrt{9-x^2}\) to \(\sqrt{9-x^2}\).
- **\(z\) bounds**: for each pair \((x, y)\), from \(0\) to \(\sqrt{9-x^2-y^2}\).
The integral evaluates the volume under the surface defined by the exponential function within the specified region, which geometrically could represent a hemisphere or similar shape suspended above the xy-plane.
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