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
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