2. Find the image of the square bounded by the lines $$ u=0, u=1, v=0, v=1 $$ under the transformation $$ x=v, y=u\left(1+v^{2}\right) $$.

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We start with the transformation given by
$$
x=v,\quad y=u(1+v^2).
$$
Since the square in the $$uv$$–plane is defined by
$$
0\leq u\leq 1,\quad 0\leq v\leq 1,
$$
we analyze the image step by step.

1. Because $$x=v$$ and $$v\in[0,1]$$, the $$x$$ coordinate in the $$xy$$–plane satisfies
   $$
   0\leq x\leq 1.
   $$

2. For a fixed value of $$v$$ (hence $$x$$), the variable $$u$$ ranges between 0 and 1. The corresponding $$y$$ coordinate is given by
   $$
   y=u(1+v^2)=u(1+x^2).
   $$
   When $$u=0$$, we have $$y=0$$; when $$u=1$$, we have
   $$
   y=1+x^2.
   $$
   Therefore, for each $$x\in[0,1]$$, $$y$$ takes all values between 0 and $$1+x^2$$.

Hence, the image of the square under the given transformation is the region
$$
\{(x,y) \mid 0\leq x\leq 1,\; 0\leq y\leq 1+x^2\}.
$$
The image of the square is the region where $$0 \leq x \leq 1$$ and $$0 \leq y \leq 1 + x^2$$.

Find the image of the square bounded by the lines under the
transformation .

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Find the image of the square bounded by the lines under the transformation .