Question

Math

Posted 4 months ago

```
Suppose we wanted to evaluate the double integral
$S=\iint_{D} \sqrt{x^{2}+y^{2}} d x d y$
by first applying a change of variables from $D$ to $R$ :
$\begin{array}{l}
x=X_{1}(u, v)=-u^{2}+v \\
y=X_{2}(u, v)=u^{2}+2 u-v
\end{array}$
What is $S$ under the change of variables?
If you know an expression within absolute value is non-negative, do not use absolute value at all.
$S=\iint_{R} \square \quad \overline{=x} d u d v$
```

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Answer from Sia

Posted 4 months ago

Solution by Steps

step 1

Calculate the Jacobian matrix for the transformation

step 2

The Jacobian matrix is given by $\begin{pmatrix} \frac{\partial X_1}{\partial u} & \frac{\partial X_1}{\partial v} \\ \frac{\partial X_2}{\partial u} & \frac{\partial X_2}{\partial v} \end{pmatrix} = \begin{pmatrix} -2u & 1 \\ 2u+2 & -1 \end{pmatrix}$

step 3

Compute the determinant of the Jacobian matrix, which is the Jacobian determinant

step 4

The determinant is $(-2u) \cdot (-1) - (1) \cdot (2u+2) = 2u - 2u - 2 = -2$

step 5

Since the determinant is negative, we take its absolute value for the integral transformation

step 6

The absolute value of the Jacobian determinant is $|-2| = 2$

step 7

Substitute the expressions for $x$ and $y$ in terms of $u$ and $v$ into the integral

step 8

The integral becomes $S = \iint_{R} \sqrt{(-u^2+v)^2 + (u^2+2u-v)^2} \cdot 2 \, du \, dv$

step 9

Simplify the integrand before integrating

step 10

The integrand simplifies to $\sqrt{u^4 - 2u^2v + v^2 + u^4 + 4u^3 - 2u^2v + 4u^2 - 4uv + v^2} \cdot 2$

step 11

Combine like terms in the integrand

step 12

The integrand further simplifies to $\sqrt{2u^4 + 4u^3 + 2u^2 + 2v^2 - 4u^2v - 4uv} \cdot 2$

step 13

Now, evaluate the integral over the region $R$

Answer

The integral $S$ under the change of variables is given by $S = \iint_{R} \sqrt{2u^4 + 4u^3 + 2u^2 + 2v^2 - 4u^2v - 4uv} \cdot 2 \, du \, dv$.

Key Concept

Change of Variables in Double Integrals

Explanation

When changing variables in a double integral, the new integral is obtained by substituting the expressions for the original variables in terms of the new variables and multiplying by the absolute value of the Jacobian determinant. The Jacobian accounts for the change in area elements due to the transformation.

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