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.
$$
S=\iint_{R} \square \quad \overline{=x} d u d v
$$

Question

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.
$$
S=\iint_{R} \square \quad \overline{=x} d u d v
$$

Accepted Answer

∻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.◊What is the expression of the Jacobian determinant for the given transformation of variables?⍭ Generate me a similar question◊⚹

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