Let R be the region in the first quadrant bounded by the line y = 5 and the graphs and y = 5 – x2. Write an integral expression for the area of R, using vertical slices.
The region R looks like this:
We found an expression for the area of R using horizontal slices earlier.
To use vertical slices, we have to split the region into two parts at x = 2. The left piece uses the graph y = 5 – x2 for its lower bound, and the right piece uses the graph of as its lower bound.
The limits of integration will be the same as in the previous problem. The left piece has 0 ≤ x ≤ 2, and the right piece has .
A slice of the left piece has height
5 – (5 – x2) = x2.
Taking the integral, the area of the left piece is
A slice of the right piece has height .
Taking the integral, the area of the right piece is
Adding the areas of the two pieces together, we conclude that the area of R is