Evaluate the iterated integral [² [² 2e3² dy dx by changing the order of integration.
Evaluate the iterated integral [² [² 2e3² dy dx by changing the order of integration.
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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![**Evaluating Iterated Integrals: Changing the Order of Integration**
In this section, we will learn how to evaluate iterated integrals by altering the order of integration. We are given a specific example to illustrate this process.
### Problem
Evaluate the iterated integral:
\[ \int_{0}^{2} \int_{x}^{2} 2e^{y^2} \, dy \, dx \]
by changing the order of integration.
### Solution Outline
1. **Understand the Region of Integration**:
- The inner integral \(\int_{x}^{2} 2e^{y^2} \, dy\) suggests that for a fixed \(x\), \(y\) ranges from \(x\) to 2.
- For the outer integral, \(x\) ranges from 0 to 2.
2. **Sketch the Region**:
- Draw the region in the \(xy\)-plane defined by \(0 \leq x \leq 2\) and \(x \leq y \leq 2\).
**Region Description**:
- The region is bounded below by \(y = x\) and above by \(y = 2\), with \(x\) ranging from 0 to 2.
3. **Reverse the Order of Integration**:
- Identify new bounds for \(x\) and \(y\) when switching the order.
- \(y\) now ranges from 0 to 2.
- For a fixed \(y\), \(x\) ranges from 0 to \(y\).
4. **Rewrite the Integral**:
- The new integral is:
\[ \int_{0}^{2} \int_{0}^{y} 2e^{y^2} \, dx \, dy \]
5. **Evaluate the New Integral**:
- Integrate the inner integral with respect to \(x\):
\[ \int_{0}^{y} 2e^{y^2} \, dx = 2e^{y^2} \times x \bigg|_{0}^{y} = 2ye^{y^2} \]
- Integrate the result with respect to \(y\):
\[ \int_{0}^{2} 2y e^{y^2} \, dy \]
- Use substitution \(u](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F3d3733a5-5e9e-433b-b6d1-2cfec636672d%2F709f2682-9d09-41c8-a275-b998bc414cd6%2Fgbdp8un_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Evaluating Iterated Integrals: Changing the Order of Integration**
In this section, we will learn how to evaluate iterated integrals by altering the order of integration. We are given a specific example to illustrate this process.
### Problem
Evaluate the iterated integral:
\[ \int_{0}^{2} \int_{x}^{2} 2e^{y^2} \, dy \, dx \]
by changing the order of integration.
### Solution Outline
1. **Understand the Region of Integration**:
- The inner integral \(\int_{x}^{2} 2e^{y^2} \, dy\) suggests that for a fixed \(x\), \(y\) ranges from \(x\) to 2.
- For the outer integral, \(x\) ranges from 0 to 2.
2. **Sketch the Region**:
- Draw the region in the \(xy\)-plane defined by \(0 \leq x \leq 2\) and \(x \leq y \leq 2\).
**Region Description**:
- The region is bounded below by \(y = x\) and above by \(y = 2\), with \(x\) ranging from 0 to 2.
3. **Reverse the Order of Integration**:
- Identify new bounds for \(x\) and \(y\) when switching the order.
- \(y\) now ranges from 0 to 2.
- For a fixed \(y\), \(x\) ranges from 0 to \(y\).
4. **Rewrite the Integral**:
- The new integral is:
\[ \int_{0}^{2} \int_{0}^{y} 2e^{y^2} \, dx \, dy \]
5. **Evaluate the New Integral**:
- Integrate the inner integral with respect to \(x\):
\[ \int_{0}^{y} 2e^{y^2} \, dx = 2e^{y^2} \times x \bigg|_{0}^{y} = 2ye^{y^2} \]
- Integrate the result with respect to \(y\):
\[ \int_{0}^{2} 2y e^{y^2} \, dy \]
- Use substitution \(u
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