# Lebesgue measure (unit square/integral)

Let $$f:[0,1]\times [0,1]\to \mathbb{R^2}, \ (x,y) \mapsto (y,x^2y+x)$$.

How to determine the Lebesgue measure of the image $$f([0,1]\times [0,1])$$?

Since it's surjective and a rectangle the Lebesgue measure should be $$2 \cdot 1=2$$, but it also ranges between $$0$$ and $$y+1$$ so the Lebesgue measure could be also $$\frac{3}{2}$$.

How to find out which of these solutions is correct?

• The map is not surjective - certainly the image of $f$ is not all of $\mathbb{R}^{2}$, since the first coordinate can only be between $0$ and $1$ – pwerth Dec 4 '18 at 23:17

You can verify that the range is exactly $$\{(x,y): 0\leq x \leq 1,0\leq y \leq x+1\}$$. The measure of this set is 1.5. If you draw a picture you can see that the set is made up of a rectangle and a triangle. So it is easy to compute the area.