Answer
(a) we show that $F$ maps ${\cal D}$ to a rectangle ${\cal R}$ in the $uv$-plane, where
${\cal R} = \left\{ {\left( {u,v} \right)|8 \le u \le 9,5 \le v \le 6} \right\}$
(b) ${\rm{Area}}\left( {\cal D} \right) \approx \frac{1}{5}$
Work Step by Step
(a) Referring to Figure 6, we see that ${\cal D}$ is defined by
${\cal D} = \left\{ {\left( {x,y} \right)|8 \le y + {x^2} \le 9,5 \le y - {x^3} \le 6} \right\}$
Since $F$ is the map: $u = y + {x^2}$, $v = y - {x^3}$, so
$8 \le u \le 9$, ${\ \ \ \ \ }$ $5 \le v \le 6$
Let ${\cal R}$ denote the rectangle defined by
${\cal R} = \left\{ {\left( {u,v} \right)|8 \le u \le 9,5 \le v \le 6} \right\}$
Therefore, $F$ maps ${\cal D}$ to a rectangle ${\cal R}$ in the $uv$-plane.
We can write $F\left( {x,y} \right) = \left( {y + {x^2},y - {x^3}} \right)$. So, the Jacobian of $F$ is
${\rm{Jac}}\left( F \right) = \left| {\begin{array}{*{20}{c}}
{\frac{{\partial u}}{{\partial x}}}&{\frac{{\partial u}}{{\partial y}}}\\
{\frac{{\partial v}}{{\partial x}}}&{\frac{{\partial v}}{{\partial y}}}
\end{array}} \right| = \left| {\begin{array}{*{20}{c}}
{2x}&1\\
{ - 3{x^2}}&1
\end{array}} \right| = 2x + 3{x^2}$
(b) Recall Eq. (7) in Section 16.6:
(1) ${\ \ \ \ }$ ${\rm{Area}}\left( {\cal D} \right) = {\rm{Area}}\left( {G\left( {\cal R} \right)} \right) \approx \left| {{\rm{Jac}}\left( G \right)} \right|{\rm{Area}}\left( {\cal R} \right)$
1. Using the definition of ${\cal R}$:
${\cal R} = \left\{ {\left( {u,v} \right)|8 \le u \le 9,5 \le v \le 6} \right\}$
we get ${\rm{Area}}\left( {\cal R} \right) = 1$.
2. Evaluate the Jacobian of $F$ at $P = \left( {1,7} \right)$:
${\rm{Jac}}{\left( F \right)_{P = \left( {1,7} \right)}} = 2 \times 1 + 3 \times {1^2} = 5$
Using Eq. (14) in Section 16.6:
${\rm{Jac}}\left( G \right) = {\rm{Jac}}{\left( F \right)^{ - 1}}$, ${\ \ \ \ \ }$ where $F = {G^{ - 1}}$
we get ${\rm{Jac}}{\left( G \right)_{P = \left( {1,7} \right)}} = \frac{1}{5}$.
Substituting these results in equation (1) gives
${\rm{Area}}\left( {\cal D} \right) \approx \frac{1}{5} \times 1 = \frac{1}{5}$
