(cos x − sin x) dx = (sin x − cos x) (sin x − cos x) dx = (−cos x − sin x)Adding these, we get that the area is 2 − 2.
HORIZONTAL SLICES
Now for the fun part. We can slice a region vertically when one function is at the top of our section and a
different function is at the bottom. But what if the same function is both the top and the bottom of the slice
(what we call a double-valued function)? You have to slice the region horizontally.
If we were to slice vertically, as in the left-hand picture, we’d have a problem. But if we were to slice
horizontally, as in the right-hand picture, we don’t have a problem. Instead of integrating an equation f(x)
with respect to x, we need to integrate an equation f(y) with respect to y. As a result, our area formula
changes a little.
If a region is bounded by f(y) on the right and g(y) on the left at all points of the interval [c, d], then
the area of the region is given by [f (y) − g(y)] dyExample 3: Find the area of the region between the curve x = y^2 and the curve x = y + 6 from y = 0 to y =