Change the order of integration in the int rl (vy f(r u) drdu
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
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![**Title: Changing the Order of Integration in Double Integrals**
**Problem Statement:**
Change the order of integration in the integral:
\[ \int_{0}^{1} \int_{y^2}^{\sqrt{y}} f(x, y) \, dx \, dy \]
**Explanation:**
The given integral is a double integral where the integration with respect to \(x\) is performed first, over the region defined by \(y^2 \leq x \leq \sqrt{y}\). The \(y\)-integration is subsequently performed for \(y\) ranging from \(0\) to \(1\).
**Steps to Change the Order of Integration:**
1. **Visualize the Region of Integration:**
- The region is bounded by \(x = y^2\) and \(x = \sqrt{y}\).
- \(y\) ranges from \(0\) to \(1\).
2. **Determine the New Limits:**
- For a fixed \(x\), \(y\) varies between \(y = x^2\) and \(y = x^{1/2}\).
- \(x\) ranges from \(0\) to \(1\).
3. **Rewrite the Integral:**
- The order of integration can be swapped:
\[ \int_{0}^{1} \int_{x^2}^{\sqrt{x}} f(x, y) \, dy \, dx \]
This results in reversing the roles of \(x\) and \(y\) in the integration process. The inner integral is now with respect to \(y\) and then integrated with respect to \(x\).
This technique is useful for simplifying integrals or making them easier to evaluate, especially when dealing with functions that are more easily integrated in one order than the other.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Ff4f46a7d-33e6-4d1b-8c9b-a4f6142b1fc8%2F4199f524-3263-49bc-bf7c-dc3bc6b6a265%2Fhvrrsrh_processed.png&w=3840&q=75)
Transcribed Image Text:**Title: Changing the Order of Integration in Double Integrals**
**Problem Statement:**
Change the order of integration in the integral:
\[ \int_{0}^{1} \int_{y^2}^{\sqrt{y}} f(x, y) \, dx \, dy \]
**Explanation:**
The given integral is a double integral where the integration with respect to \(x\) is performed first, over the region defined by \(y^2 \leq x \leq \sqrt{y}\). The \(y\)-integration is subsequently performed for \(y\) ranging from \(0\) to \(1\).
**Steps to Change the Order of Integration:**
1. **Visualize the Region of Integration:**
- The region is bounded by \(x = y^2\) and \(x = \sqrt{y}\).
- \(y\) ranges from \(0\) to \(1\).
2. **Determine the New Limits:**
- For a fixed \(x\), \(y\) varies between \(y = x^2\) and \(y = x^{1/2}\).
- \(x\) ranges from \(0\) to \(1\).
3. **Rewrite the Integral:**
- The order of integration can be swapped:
\[ \int_{0}^{1} \int_{x^2}^{\sqrt{x}} f(x, y) \, dy \, dx \]
This results in reversing the roles of \(x\) and \(y\) in the integration process. The inner integral is now with respect to \(y\) and then integrated with respect to \(x\).
This technique is useful for simplifying integrals or making them easier to evaluate, especially when dealing with functions that are more easily integrated in one order than the other.
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