Question

Evaluate the given integral by making an appropriate change of variables.

Answer 1

The integrand and given boundary for R suggests we should use

u = x - 4y

v = 5x - y

with 0 ≤ u ≤ 3 and 6 ≤ v ≤ 9.

Then

x = (4v - u)/19

y = (v - 5u)/19

and the Jacobian for this transformation is

`J=\begin{bmatrix}x_u&x_v\\y_u&y_v\end{bmatrix}=\frac1{19}\begin{bmatrix}-1&4\\-5&1\end{bmatrix}\implies|\det J|=\left|\frac1{19}\right|=\frac1{19}`

The integral is then

`\displaystyle\iint_R 2(x-4y)/(5x-y)\,\mathrm dA = \frac1{19} \int_6^9 \int_0^3 \frac{2u}v\,\mathrm du\,\mathrm dv`

`=\displaystyle \frac2{19} \int_6^9 (3^2-0^2)/(2v)\,\mathrm dv`

`=\displaystyle \frac9{19} \int_6^9 \frac{\mathrm dv}v`

`=\frac9{19}(\ln9-\ln6)=\frac9{19}\ln\frac96=\boxed{\frac9{19}\ln\frac32}`