Let R be the region bounded by the graphs of y = ex, y = 1, and x = 3.
Write an integral expression for the area of R, using
(a) vertical slices
(b) horizontal slices
The region R looks like this:
Within the region R, x goes from 0 to 3 and y goes from 1 to e3. These will be our limits of integration later.
(a) A vertical slice of R looks like this:
The height of the slice at position x is (ex – 1), so the area of the slice is (ex – 1) Δ x. Using the limits of integration we found earlier,
the area of R is
(b) A horizontal slice of R looks like this:
The width of the slice at height y is
3 – x = 3 – ln y
so the area of the slice is
(3 – ln y) Δ y.
Using the limits of integration we found already, the area of R is