Let *R* be the region in the first quadrant bounded by the *x*-axis and the graphs and *y* = 5 – *x*^{2}. Write an integral expression for the area of *R*, using vertical slices.

Answer

The region *R* looks like this:

Earlier, we wrote an expression for the area of *R* using horizontal slices.

To use vertical slices, we have to split *R* into two pieces. The pieces on the left is bounded by the graph of , the line *x* = 2, and the *x*-axis. The piece on the right is bounded by the graph *y* = 5 – *x*^{2}, the line *x* = 2, and the *x*-axis. Let's look at the piece on the left first. The slice at position *x* has height , and *x* runs from 0 to 2. The area of the piece on the left is

For the piece on the right, the slice at position *x* has height 5 – *x*^{2}, and it *x* runs from 2 to (since this is where the graph of 5 – *x*^{2} crosses the *x*-axis).

The area of the piece on the right is

Adding up the areas of the two pieces, the area of *R* is