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 ⋮ To evaluate the double integral with a change of variables, we first need to find the Jacobian determinant of the transformation. ‖ ‖ step 2 ⋮ The Jacobian matrix is given by $\begin{bmatrix} \frac{\partial x}{\partial u} & \frac{\partial x}{\partial v} \ \frac{\partial y}{\partial u} & \frac{\partial y}{\partial v} \end{bmatrix}$, which for our transformation is $\begin{bmatrix} -2u & 1 \ 2u+2 & -1 \end{bmatrix}$. ‖ ‖ step 3 ⋮ The Jacobian determinant, $J(u,v)$, is the determinant of the Jacobian matrix: $J(u,v) = (-2u)(-1) - (1)(2u+2)$. ‖ ‖ step 4 ⋮ Simplifying the Jacobian determinant gives us $J(u,v) = 2u - 2u - 2 = -2$. ‖ ‖ step 5 ⋮ Since we know that $J(u,v)$ is non-negative, we can omit the absolute value and use $J(u,v) = 2$ for the change of variables. ‖ ‖ step 6 ⋮ The double integral $S$ under the change of variables is then $S = \iint_{R} \sqrt{(-u^2+v)^2+(u^2+2u-v)^2} \cdot |J(u,v)| \, du \, dv$. ‖ ‖ step 7 ⋮ Substituting the Jacobian determinant, we get $S = \iint_{R} \sqrt{(-u^2+v)^2+(u^2+2u-v)^2} \cdot 2 \, du \, dv$. ‖ ‖ step 8 ⋮ Simplify the integrand to find the expression in terms of $u$ and $v$. ‖ ∻Answer∻ ⚹ The expression for $S$ under the change of variables is $S = \iint_{R} \sqrt{(-u^2+v)^2+(u^2+2u-v)^2} \cdot 2 \, du \, dv$. ⚹ ∻Key Concept∻ ⚹Change of Variables in Double Integrals⚹ ∻Explanation∻ ⚹When changing variables in a double integral, the integrand and the differential area element $dx \, dy$ are transformed using the new variables and the absolute value of the Jacobian determinant of the transformation.⚹◊What is the expression for the Jacobian determinant when performing the change of variables in the double integral presented by the student?⍭ Generate me a similar question◊

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