region of integration
Every multiple integral lives over some set — a patch of the plane for a double integral, a solid chunk of space for a triple integral. That set is the region of integration. Before you can write down a single limit, you have to know the shape you are summing over: a disk, a triangle, the area between two parabolas, a cylinder capped by a paraboloid. Half the difficulty of multiple integration is not the integrand at all — it is describing this region precisely enough to pin down the limits.
Regions come in convenient flavors. A region is called type I (or vertically simple) if it sits between two curves y = g(x) and y = h(x) over an x-interval [a, b]; then a vertical line at any x enters at g(x) and exits at h(x), giving the inner limits for dy dx. It is type II (horizontally simple) if it sits between x = p(y) and x = q(y) over a y-interval; a horizontal line gives the inner limits for dx dy. Many regions are both, and you pick whichever makes the integral easier. A region that is neither simple type gets carved into a few pieces that each are, and the integral is the sum over the pieces. Crucially, the geometry of the region — not the integrand — sets every limit of integration.
Reading the region correctly is where most setup errors are born and where most of them can be cured. The discipline is always the same: sketch it, find the boundary curves and where they intersect, decide the order, then for the inner variable trace a sweeping line and record where it enters and leaves. The region also dictates whether you should change coordinates: a disk or annulus screams polar, a sphere screams spherical, a cylinder screams cylindrical — choosing coordinates that match the region's symmetry usually turns ugly variable limits into plain constants.
The region between y = x^2 and y = x (which meet at (0,0) and (1,1)) is type I: for each x in [0,1], y runs from x^2 up to x. So integral over R of f dA = integral from 0 to 1 of [integral from x^2 to x of f dy] dx.
Finding where the boundary curves intersect fixes the outer constant limits; the curves themselves are the inner limits.
The integrand never sets the limits — the region does. A common error is letting an x or y appearing in f leak into the bounds; the bounds come only from the boundary curves of R.