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Evaluate the following triple integral by hand: 2 }}} ij X'√= dxdydz= y³ z=1 y=2 x=-1

Evaluate the following triple integral by hand: 2 }}} ij X'√= dxdydz= y³ z=1 y=2 x=-1

Calculus For The Life Sciences
Calculus For The Life Sciences
2nd Edition
ISBN:9780321964038
Author:GREENWELL, Raymond N., RITCHEY, Nathan P., Lial, Margaret L.
Publisher:GREENWELL, Raymond N., RITCHEY, Nathan P., Lial, Margaret L.
Chapter8: Further Techniques And Applications Of Integration
Section8.CR: Chapter 8 Review
Problem 12CR
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### Evaluating a Triple Integral by Hand In this exercise, we are required to evaluate the following triple integral by hand: \[ \int_{z=1}^{9} \int_{y=2}^{4} \int_{x=-1}^{2} \frac{x^4 \sqrt{z}}{y^3} \, dx \, dy \, dz \] #### Steps to Solve: 1. **Set up the Integral:** - The integral is set up in the standard order of integration from inner to outer: \(dx\), \(dy\), \(dz\). 2. **Integrate with respect to \(x\):** - Evaluate the innermost integral (with respect to \(x\)) while treating \(y\) and \(z\) as constants: \[ \int_{x=-1}^{2} \frac{x^4 \sqrt{z}}{y^3} \, dx \] 3. **Integrate with respect to \(y\):** - After simplifying the result from \(x\) integration, integrate with respect to \(y\): \[ \int_{y=2}^{4} \left( \text{Result from the } x \text{-integration} \right) \, dy \] 4. **Integrate with respect to \(z\):** - Finally, integrate with respect to \(z\) using the result from the \(y\) integration: \[ \int_{z=1}^{9} \left( \text{Result from the } y \text{-integration} \right) \, dz \] By following these steps in sequence, one can evaluate the given triple integral manually.
Transcribed Image Text:### Evaluating a Triple Integral by Hand In this exercise, we are required to evaluate the following triple integral by hand: \[ \int_{z=1}^{9} \int_{y=2}^{4} \int_{x=-1}^{2} \frac{x^4 \sqrt{z}}{y^3} \, dx \, dy \, dz \] #### Steps to Solve: 1. **Set up the Integral:** - The integral is set up in the standard order of integration from inner to outer: \(dx\), \(dy\), \(dz\). 2. **Integrate with respect to \(x\):** - Evaluate the innermost integral (with respect to \(x\)) while treating \(y\) and \(z\) as constants: \[ \int_{x=-1}^{2} \frac{x^4 \sqrt{z}}{y^3} \, dx \] 3. **Integrate with respect to \(y\):** - After simplifying the result from \(x\) integration, integrate with respect to \(y\): \[ \int_{y=2}^{4} \left( \text{Result from the } x \text{-integration} \right) \, dy \] 4. **Integrate with respect to \(z\):** - Finally, integrate with respect to \(z\) using the result from the \(y\) integration: \[ \int_{z=1}^{9} \left( \text{Result from the } y \text{-integration} \right) \, dz \] By following these steps in sequence, one can evaluate the given triple integral manually.
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