The area in 1) is not bounded by the three relations. So it cannot be calculated, and is in any case not equal to $J(b)$.
Edit: Now that the conditions in (1) has been corrected, it is possible to determine whether or not it is true. And it is, given some assumption like the one in my comment below, namely that if $f(0) = c > 0$, then we set $f^{-1}(y) = 0$ for all non-negative $yEnd edit
If $b = f(a)$, then $I(a)$ and $J(b)$ will be the area of two parts of the rectangle with corners $(0, 0), (a, 0), (a, b), (b, 0)$, so in that case $I(a) + J(b) = ab$.
(2) is false, and (3) is true, since if $b\neq f(a)$, then $ab$ will be smaller than the areas of the two integrals. You really should draw a picture of the whole setting, just to convince yourself of this.
A written reasoning for the above paragraph can be done as follows: Assume $b ab$