Let R be the region above the graph of y = 1 – x2 and below the graph x2 + y2 = 1. Write an integral expression for the area of R, using vertical slices.
The region R looks like this:
Since R is symmetric across the y-axis, we can find the area of the part that's in quadrant 1 and then multiply by 2 to find the full area of R.
The vertical slice of R at position x looks like this:
The height of this slice is
and the area is
As x can go from 0 to 1, the area of the right half of R is
and the area of all of R is
If you wanted to find the area of all of R at once, x would go from -1 to 1 and the area of R would be