Let the four successively larger square have sides, x, x+1, x+2, x+3, then the square large than 1 must have side 4. that means the upper right square has side x+7 and the lower right square has side x+11. Furthermore, the lower left square is x+11+4-x, i.e., 15; so, x must be 7.

This means the base is 15+18=33, and the height is 18+14=32. Therefore the area of the rectangle is 1056.

Cross checking: 1+16+49+64+81+100+14*14+18*18+15*15=90

Disclosure: this took me more than 100 seconds!