Change of Variables in Double Integrals Examples 2

# Change of Variables in Double Integrals Examples 2

Recall from the Change of Variables in Double Integrals page that if $z = f(x, y)$ is a continuous two variable real-valued function, $T : S \to R$ is a one-to-one transformation (except possibly on the boundary of $S$), the Jacobian Determinant $\frac{\partial (x, y)}{\partial (u, v)}$ is nonzero, and $x = x(u, v)$, $y = y(u, v)$, and their first partial derivatives with respect to $u$ and $v$ are continuous, then:

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\begin{align} \quad \iint_R f(x, y) \: dA = \iint_S f(x(u, v), y(u, v)) \biggr \rvert \frac{\partial (x, y)}{\partial (u, v)} \biggr \rvert \: du \: dv \end{align}

We will now look at some more examples of evaluating double integrals using a change of variables.

## Example 1

Evaluate $\iint_D xy \: dA$ where $D$ is the region in the first quadrant bounded by the curves $y = x$, $y = 3x$, $y = \frac{1}{x}$, $y = \frac{3}{x}$ and using the transformation $x = \frac{u}{v}$ and $y = v$.

The region that we wish to integrate over is given below:

We need to apply our transformation to each of the boundary curves. We will have that $y = x$ implies that $v^2 = u$, $y = 3x$ implies that $v = 3 \frac{u}{v}$ which implies that $v^2 = 3u$, $y = \frac{1}{x}$ implies that $v = \frac{v}{u}$ which implies that $u = 1$, and $y = \frac{3}{x}$ implies that $v = 3 \frac{v}{u}$ which implies that $u = 3$.

So our new boundary curves are $v^2 = u$, $v^2 = 3u$, $u = 1$ and $u = 3$:

This region, call it $D_{uv}$ can be described as:

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\begin{align} \quad D_{uv} = \{ (u, v) : 1 ≤ u ≤ 3, \sqrt{u} ≤ v ≤ \sqrt{3u} \} \end{align}

We now need to compute the Jacobian for this transformation:

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\begin{align} \quad \frac{\partial (x, y)}{\partial (u, v)} = \begin{vmatrix} \frac{\partial x}{\partial u} & \frac{\partial x}{\partial v} \\ \frac{\partial y}{\partial u} & \frac{\partial y}{\partial v} \end{vmatrix} = \begin{vmatrix} \frac{1}{v} & - \frac{u}{v^2} \\ 0 & 1 \end{vmatrix} = \frac{1}{v} \end{align}

We are now ready to compute this integral.

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