Use the change of variables s = xy, t = xy² to compute xy² dA, where R is the region bounded by xy = 4, xy = 5, xy² = 4, xy² = 5. SRxy² dA=

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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 \(
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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