Calculate the double integral \[\iint\limits_R {dxdy},\] where the region \(R\) is bounded by the parabolas \({y^2} = 2x,\) \({y^2} = 3x\) and hyperbolas \(xy = 1,\) \(xy = 2.\)
Solution.
The region \(R\) is sketched in Figure \(5.\)
Figure 5.
We apply the following substitution of variables to simplify the region \(R:\)
\[
\left\{ \begin{array}{l}
u = \frac{{{y^2}}}{x}\\
v = xy
\end{array} \right..\]
The pullback \(S\) of the region \(R\) is defined as follows:
As it can be seen, the region \(S\) is the rectangle. To find the Jacobian of the transformation, we express the variables \(x, y\) in terms of \(u, v.\)
\[u = \frac{{{y^2}}}{x},\;\; \Rightarrow x = \frac{{{y^2}}}{u},\]
Evaluate the integral \[\iint\limits_R {\left( {{x^2} + {y^2}} \right)dxdy},\] where \(R\) is bounded by the lines
\[y = x, y = x + a, y = a, y = 2a \left(a \gt 0\right).\]
Solution.
The region of integration \(R\) is a parallelogram and is shown in Figure \(6.\)
Figure 6.
We can make the following change of variables:
\[
\left\{ \begin{array}{l}
u = y - x\\
v = y
\end{array} \right.\;\;
\text{or}\;\;\left\{ \begin{array}{l}
x = y - u = v - u\\
y = v
\end{array} \right..\]
The purpose of this change is to simplify shape of the region of integration \(R.\)
The image \(S\) of \(R\) in terms of \(\left( {u,v} \right)\) is defined as
\[y = x,\;\; \Rightarrow y - x = 0,\;\; \Rightarrow u = 0,\]
\[y = x + a,\;\; \Rightarrow y - x = a,\;\; \Rightarrow u = a,\]
\[y = a,\;\; \Rightarrow v = a,\]
\[y = 2a,\;\; \Rightarrow v = 2a.\]
As it can be seen from the Figure \(7,\) \(S\) is the rectangular region.
