Calculus: Early Transcendentals
8th Edition
ISBN:9781285741550
Author:James Stewart
Publisher:James Stewart
Chapter1: Functions And Models
Section: Chapter Questions
Problem 1RCC: (a) What is a function? What are its domain and range? (b) What is the graph of a function? (c) How...
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![**Evaluating the Integral by Changing the Order of Integration**
To evaluate the integral by changing the order of integration in an appropriate way, we start with the given integral:
\[ \int_0^1 \int_{\frac{1}{2z}}^1 \int_0^{\ln 8} \frac{\pi e^{2x} \sin\left(\frac{\pi y}{2}\right)}{y^2} \, dx \, dy \, dz \]
To solve this, we must carefully analyze each integration limit and consider changing the order of integration.
1. **Original Integral Expression:**
\[ \int_0^1 \int_{\frac{1}{2z}}^1 \int_0^{\ln 8} \frac{\pi e^{2x} \sin\left(\frac{\pi y}{2}\right)}{y^2} \, dx \, dy \, dz \]
2. **Steps to Change Order of Integration:**
- **Identify the Bounds:**
- The outermost integral \(\int_0^1 dz\) indicates \(z\) ranges from 0 to 1.
- The middle integral \(\int_{\frac{1}{2z}}^1 dy \) indicates \(y\) ranges from \( \frac{1}{2z} \) to 1.
- The innermost integral \(\int_0^{\ln 8} dx\) indicates \(x\) ranges from 0 to \(\ln 8\).
- **Determine the Region of Integration:**
- For \(z\): \(0 \leq z \leq 1\).
- For \(y\): \(\frac{1}{2z} \leq y \leq 1\).
- For \(x\): \(0 \leq x \leq \ln 8\).
- Consider the difficulties arising from the term \(\frac{1}{2z} \leq y \leq 1\).
- To simplify the region, switch the variables of integration.
3. **New Order of Integration:**
Switch the order to \( y \)-integration first, followed by \( z \)-integration, and then \( x \)-integration. The new integral setup involves the following bounds:
- For \(](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fb21cc1b3-0a4b-474e-87d5-2a9f16683342%2F022ef008-a4ce-4215-8053-232265c2a6bc%2Fq5ydqw_processed.png&w=3840&q=75)
Transcribed Image Text:**Evaluating the Integral by Changing the Order of Integration**
To evaluate the integral by changing the order of integration in an appropriate way, we start with the given integral:
\[ \int_0^1 \int_{\frac{1}{2z}}^1 \int_0^{\ln 8} \frac{\pi e^{2x} \sin\left(\frac{\pi y}{2}\right)}{y^2} \, dx \, dy \, dz \]
To solve this, we must carefully analyze each integration limit and consider changing the order of integration.
1. **Original Integral Expression:**
\[ \int_0^1 \int_{\frac{1}{2z}}^1 \int_0^{\ln 8} \frac{\pi e^{2x} \sin\left(\frac{\pi y}{2}\right)}{y^2} \, dx \, dy \, dz \]
2. **Steps to Change Order of Integration:**
- **Identify the Bounds:**
- The outermost integral \(\int_0^1 dz\) indicates \(z\) ranges from 0 to 1.
- The middle integral \(\int_{\frac{1}{2z}}^1 dy \) indicates \(y\) ranges from \( \frac{1}{2z} \) to 1.
- The innermost integral \(\int_0^{\ln 8} dx\) indicates \(x\) ranges from 0 to \(\ln 8\).
- **Determine the Region of Integration:**
- For \(z\): \(0 \leq z \leq 1\).
- For \(y\): \(\frac{1}{2z} \leq y \leq 1\).
- For \(x\): \(0 \leq x \leq \ln 8\).
- Consider the difficulties arising from the term \(\frac{1}{2z} \leq y \leq 1\).
- To simplify the region, switch the variables of integration.
3. **New Order of Integration:**
Switch the order to \( y \)-integration first, followed by \( z \)-integration, and then \( x \)-integration. The new integral setup involves the following bounds:
- For \(
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