General change of coordinates

1 General changes of coordinates

Using an appropriate change of variables, compute
\[ \iint _D x^2 y \, dA \]
where \(D \subset \mathbb {R}^2\) is the region between the hyperbolas \(y = 2 / x\) and \(y = 4 /x\) and the lines \(y= 1\) and \(y = 5\)
The region can be described as
\begin{align*} 2 &\leq xy \leq 4 \\ 1 & \leq y \leq 5 \end{align*}
Using an appropriate change of variables, compute
\[ \iint _D x^2 \, dA \]
where \(D \subset \mathbb {R}^2\) is the region between the hyperbolas \(y = 2 / x\) and \(y = 3 /x\) and the lines \(y= x\) and \(y = 4x\)
The region can be described as
\begin{align*} 2 &\leq xy \leq 3 \\ 1 & \leq y/x \leq 4 \end{align*}
Using an appropriate change of variables, compute
\[ \iint _D 3x \, dA \]
where \(D \subset \mathbb {R}^2\) is the parallelogram with vertices \((2,0)\), \((4,3)\), \((3,4)\), \((1,1)\).
The sides of the parallogram have slopes \(3/2\) and \(- 1\)
Using an appropriate change of variables, compute
\[ \iint _D \frac {y}{3y + x} \, dA \]
where \(D \subset \mathbb {R}^2\) is the parallelogram with vertices \((-1,2)\), \((2,1)\), \((3,3)\), \((0,4)\).
The sides of the parallogram have slopes \(2\) and \(- 1/3\)
Using an appropriate change of variables, compute
\[ \iint _D (x+y) \, dA \]
where \(D \subset \mathbb {R}^2\) is the region bounded by
  • the lines \(y = -x\) and \(y = -x + 2\)
  • the portion of the parabola \(y = x^2 \) with \(x \geq 0\)
  • the portion of the parabola \(y = x (x-2)\) with \(x \geq 1\)
If
\begin{align*} x & = u + v \\ y & = u^2 - v \end{align*}

then the region can be described as

\begin{align*} 0 &\leq u \leq 1 \\ 0 & \leq v \leq 2 \end{align*}