Let *R* be the region bounded by the graphs of *y* = *e*^{x}, *y* = 1, and *x* = 3.

Write an integral expression for the area of *R*, using

(a) vertical slices

(b) horizontal slices

Answer

The region *R* looks like this:

Within the region *R*, *x* goes from 0 to 3 and *y* goes from 1 to *e*^{3}. 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 (*e*^{x} – 1), so the area of the slice is (*e*^{x} – 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