Calculus: Early Transcendentals
8th Edition
ISBN: 9781285741550
Author: James Stewart
Publisher: Cengage Learning
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![**Problem 21: Volume of a Solid Using Double Integrals**
**Objective:**
Calculate the volume of the solid bounded above by the surface defined by the equation \( z = xye^{xy^2} \) and below by a rectangular region in the xy-plane.
**Given Parameters:**
- Surface function: \( z = xye^{xy^2} \)
- Rectangular region \( R \):
- \( 0 \leq x \leq 2 \)
- \( 0 \leq y \leq 1 \)
**Instructions:**
To find the volume of the solid, set up and evaluate a double integral over the region \( R \). The integral will be of the form:
\[
\int_0^2 \int_0^1 xye^{xy^2} \, dy \, dx
\]
**Approach:**
1. **Set Up the Double Integral:**
- Identify the limits of integration for \( x \) and \( y \) based on the given rectangle \( R \).
- Write out the integral for \( z = xye^{xy^2} \) over this region.
2. **Evaluate the Integral:**
- Compute the inner integral with respect to \( y \) while treating \( x \) as a constant.
- Compute the resulting outer integral with respect to \( x \).
This process will yield the volume of the solid within the specified boundaries. Make use of integration techniques such as substitution or integration by parts if necessary.](https://content.bartleby.com/qna-images/question/c6389447-1237-4af0-b5c6-eb1260425b55/31d5f952-d9af-4a71-8550-91b89e0168b7/gcpdosf_processed.png)
Transcribed Image Text:**Problem 21: Volume of a Solid Using Double Integrals**
**Objective:**
Calculate the volume of the solid bounded above by the surface defined by the equation \( z = xye^{xy^2} \) and below by a rectangular region in the xy-plane.
**Given Parameters:**
- Surface function: \( z = xye^{xy^2} \)
- Rectangular region \( R \):
- \( 0 \leq x \leq 2 \)
- \( 0 \leq y \leq 1 \)
**Instructions:**
To find the volume of the solid, set up and evaluate a double integral over the region \( R \). The integral will be of the form:
\[
\int_0^2 \int_0^1 xye^{xy^2} \, dy \, dx
\]
**Approach:**
1. **Set Up the Double Integral:**
- Identify the limits of integration for \( x \) and \( y \) based on the given rectangle \( R \).
- Write out the integral for \( z = xye^{xy^2} \) over this region.
2. **Evaluate the Integral:**
- Compute the inner integral with respect to \( y \) while treating \( x \) as a constant.
- Compute the resulting outer integral with respect to \( x \).
This process will yield the volume of the solid within the specified boundaries. Make use of integration techniques such as substitution or integration by parts if necessary.
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