Advanced Engineering Mathematics
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ISBN: 9780470458365
Author: Erwin Kreyszig
Publisher: Wiley, John & Sons, Incorporated
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![### Computation of Integral Using Change of Variables
In this problem, we are required to compute the integral
\[
\int_R xy^2 \, dA
\]
using a change of variables.
#### Step-by-Step Solution
1. **Change of Variables:**
Let us use the following change of variables:
\[
s = xy
\]
\[
t = xy^2
\]
2. **Region \( R \):**
The region \( R \) is bounded by the equations:
\[
xy = 4, \quad xy = 5, \quad xy^2 = 4, \quad xy^2 = 5.
\]
3. **Transformation:**
Rewrite the given boundaries in terms of the new variables \( s \) and \( t \).
- \( xy = 4 \) becomes \( s = 4 \)
- \( xy = 5 \) becomes \( s = 5 \)
- \( xy^2 = 4 \) becomes \( t = 4 \)
- \( xy^2 = 5 \) becomes \( t = 5 \)
4. **Bounds in \( s \) and \( t \):**
The region \( R \) in the \( (s, t) \)-plane is bounded as follows:
\[
4 \leq s \leq 5 \quad \text{and} \quad 4 \leq t \leq 5.
\]
5. **Jacobian Determinant:**
Compute the Jacobian determinant \( \frac{\partial (x, y)}{\partial (s, t)} \) of the transformation. In case of any specific relationships to the Jacobian as per the given transformation, adjust accordingly.
6. **Integral Evaluation:**
Transform the integral into the new variables and evaluate over the new region.
\[
\int_R xy^2 \, dA = \int \int_{(s, t) \in R'} \text{(transformed integrand)} \, \, \text{(Jacobian determinant)} \, ds \, dt
\]
where \( R' \) denotes the region in the new variables.
Finally, solve the integral in the new context defined by \(](https://content.bartleby.com/qna-images/question/bbff2935-77bb-4550-bfd1-d595e6271f30/96f276bc-07fb-4df4-9693-48230f5d054d/f91pg49_processed.png)
Transcribed Image Text:### Computation of Integral Using Change of Variables
In this problem, we are required to compute the integral
\[
\int_R xy^2 \, dA
\]
using a change of variables.
#### Step-by-Step Solution
1. **Change of Variables:**
Let us use the following change of variables:
\[
s = xy
\]
\[
t = xy^2
\]
2. **Region \( R \):**
The region \( R \) is bounded by the equations:
\[
xy = 4, \quad xy = 5, \quad xy^2 = 4, \quad xy^2 = 5.
\]
3. **Transformation:**
Rewrite the given boundaries in terms of the new variables \( s \) and \( t \).
- \( xy = 4 \) becomes \( s = 4 \)
- \( xy = 5 \) becomes \( s = 5 \)
- \( xy^2 = 4 \) becomes \( t = 4 \)
- \( xy^2 = 5 \) becomes \( t = 5 \)
4. **Bounds in \( s \) and \( t \):**
The region \( R \) in the \( (s, t) \)-plane is bounded as follows:
\[
4 \leq s \leq 5 \quad \text{and} \quad 4 \leq t \leq 5.
\]
5. **Jacobian Determinant:**
Compute the Jacobian determinant \( \frac{\partial (x, y)}{\partial (s, t)} \) of the transformation. In case of any specific relationships to the Jacobian as per the given transformation, adjust accordingly.
6. **Integral Evaluation:**
Transform the integral into the new variables and evaluate over the new region.
\[
\int_R xy^2 \, dA = \int \int_{(s, t) \in R'} \text{(transformed integrand)} \, \, \text{(Jacobian determinant)} \, ds \, dt
\]
where \( R' \) denotes the region in the new variables.
Finally, solve the integral in the new context defined by \(
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