Let R be the region in the first quadrant bounded by the x-axis and the graphs and y = 5 – x2. Write an integral expression for the area of R, using horizontal slices.
The region R looks like this:
The slice at height y stretches from the graph of (or x = 2y) on the left to the graph of y = 5 – x2 (or ) on the right:
The width of this slice is , and the area of the slice is
In the region R, the variable y ranges from 0 to 1:
Taking the integral, the area of R is