Advanced Engineering Mathematics
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ISBN: 9780470458365
Author: Erwin Kreyszig
Publisher: Wiley, John & Sons, Incorporated
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![**Title:** Evaluating Triple Integrals with Different Orders of Integration
**Objective:** Learn how to evaluate a triple integral by applying three different orders of integration.
**Problem:** Evaluate the integral using three different orders of integration.
\[
\iiint\limits_{E} (xy + z^2) \, dV
\]
where the region \( E \) is defined as:
\[
E = \{(x, y, z) \,|\, 0 \leq x \leq 2, \, 0 \leq y \leq 1, \, 0 \leq z \leq 3 \}
\]
**Approach:**
The integral needs to be evaluated by integrating the function \( xy + z^2 \) over the region \( E \) using three different integration orders. The possible orders of integration are:
1. **Order 1:** \( \int_{0}^{3} \int_{0}^{1} \int_{0}^{2} (xy + z^2) \, dx \, dy \, dz \)
2. **Order 2:** \( \int_{0}^{2} \int_{0}^{3} \int_{0}^{1} (xy + z^2) \, dy \, dz \, dx \)
3. **Order 3:** \( \int_{0}^{1} \int_{0}^{2} \int_{0}^{3} (xy + z^2) \, dz \, dx \, dy \)
**Conclusion:**
By computing these integrals, you will understand how changing the order of integration can affect the process of solving a triple integral, although the final result should remain consistent across all orders.](https://content.bartleby.com/qna-images/question/251d1f23-479f-46f1-ad40-8e7a3447a718/7b2d3a41-c8e9-404b-ac45-5840ec64164b/glynb8r_processed.png)
Transcribed Image Text:**Title:** Evaluating Triple Integrals with Different Orders of Integration
**Objective:** Learn how to evaluate a triple integral by applying three different orders of integration.
**Problem:** Evaluate the integral using three different orders of integration.
\[
\iiint\limits_{E} (xy + z^2) \, dV
\]
where the region \( E \) is defined as:
\[
E = \{(x, y, z) \,|\, 0 \leq x \leq 2, \, 0 \leq y \leq 1, \, 0 \leq z \leq 3 \}
\]
**Approach:**
The integral needs to be evaluated by integrating the function \( xy + z^2 \) over the region \( E \) using three different integration orders. The possible orders of integration are:
1. **Order 1:** \( \int_{0}^{3} \int_{0}^{1} \int_{0}^{2} (xy + z^2) \, dx \, dy \, dz \)
2. **Order 2:** \( \int_{0}^{2} \int_{0}^{3} \int_{0}^{1} (xy + z^2) \, dy \, dz \, dx \)
3. **Order 3:** \( \int_{0}^{1} \int_{0}^{2} \int_{0}^{3} (xy + z^2) \, dz \, dx \, dy \)
**Conclusion:**
By computing these integrals, you will understand how changing the order of integration can affect the process of solving a triple integral, although the final result should remain consistent across all orders.
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