V9 - y2 xz dz dx dy -V 9 – y2 'V x² + y2
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Evaluate the
![The image shows a triple integral set up in Cartesian coordinates. The integral is structured as follows:
\[
\int_{-3}^{3} \int_{-\sqrt{9-y^2}}^{\sqrt{9-y^2}} \int_{\sqrt{x^2+y^2}}^{4} xz \, dz \, dx \, dy
\]
### Explanation:
1. **Outer Integral (\(dy\))**: The outermost integral has limits from -3 to 3. This range is along the y-axis.
2. **Middle Integral (\(dx\))**: The middle integral is with respect to \(x\), with limits from \(-\sqrt{9-y^2}\) to \(\sqrt{9-y^2}\). These bounds describe a circle when considering the \(x\) and \(y\) axes, as they define \(x\) in terms of \(y\).
3. **Inner Integral (\(dz\))**: The innermost integral is with respect to \(z\), with the lower limit \(\sqrt{x^2+y^2}\) and the upper limit 4. This indicates a region above the volume of a cylinder (considering that \(\sqrt{x^2+y^2}\) is the radius in the xy-plane).
4. **Integrand (\(xz\))**: The function being integrated over the region is \(xz\), which implies calculating the weighted volume where the weight is a function of \(x\) and \(z\).
This integration problem typically represents a volume above a circular base, between two surfaces, or within some circular constraint in three-dimensional space.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F4776c488-bd56-43d7-a281-357c0247fbc6%2Fe228e063-bd33-4550-929a-3eaaa5c1aa63%2Fn7qe7bn_processed.png&w=3840&q=75)
Transcribed Image Text:The image shows a triple integral set up in Cartesian coordinates. The integral is structured as follows:
\[
\int_{-3}^{3} \int_{-\sqrt{9-y^2}}^{\sqrt{9-y^2}} \int_{\sqrt{x^2+y^2}}^{4} xz \, dz \, dx \, dy
\]
### Explanation:
1. **Outer Integral (\(dy\))**: The outermost integral has limits from -3 to 3. This range is along the y-axis.
2. **Middle Integral (\(dx\))**: The middle integral is with respect to \(x\), with limits from \(-\sqrt{9-y^2}\) to \(\sqrt{9-y^2}\). These bounds describe a circle when considering the \(x\) and \(y\) axes, as they define \(x\) in terms of \(y\).
3. **Inner Integral (\(dz\))**: The innermost integral is with respect to \(z\), with the lower limit \(\sqrt{x^2+y^2}\) and the upper limit 4. This indicates a region above the volume of a cylinder (considering that \(\sqrt{x^2+y^2}\) is the radius in the xy-plane).
4. **Integrand (\(xz\))**: The function being integrated over the region is \(xz\), which implies calculating the weighted volume where the weight is a function of \(x\) and \(z\).
This integration problem typically represents a volume above a circular base, between two surfaces, or within some circular constraint in three-dimensional space.
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