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

2x = 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