Let *R* be the region above the line and below the graph of *y* = cos *x* on the interval

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

(a) vertical slices

(b) horizontal slices

Answer

First graph *y* = cos *x* and :

Then shade in the region between these two graphs on the interval . This is the region *R*:

(a) With vertical slices, we have slices of width Δ *x*. Since the graph of *y* = cos *x* is above the graph of on this interval, the height of the slice at position *x* is

This means the area of a slice is

We were told to look at the interval . The integral expression

gives the area of *R*. Alternately, since the region *R* is symmetric with the *y*-axis as its line of symmetry, we could find the area of half of *R* and then multiply by 2. The expression

also gives the area of *R*.

(b) A horizontal slice of *R* looks like this:

The thickness of a slice is Δ *y*. The width of the slice at height *y* is approximately twice *x*, where (*x*,*y*) is the point on the graph of *y* = cos *x* that lies at the right edge of the slice.

If *y* = cos *x* then *x* = cos^{-1} *y*, so the width of a slice is

2*x* = 2cos^{-1} *y*.

The area of the slice is then

2cos^{-1} *y* Δ *y*.

The values of *y* that make sense are from to *y* = 1.

The area of *R* is