Let R be the region bounded by the curve y = x4 and the lines x = 2 and y = 81. Write an integral expression for the area of R, using
(a) vertical slices
(b) horizontal slices
The region R looks like this:
The curve y = x4 intersects x = 2 at the point (2,16) and intersects the line y = 81 at the point (3,81). Knowing these points will be useful for figuring out limits of integration.
(a) A vertical slice of R looks like this:
The height of the slice at position x is 81 – x4.
The area of this slice is
(81 – x4) Δ x.
The values of x that make sense are 2 ≤ x ≤ 3.
So the area of R is
(b) A horizontal slice of R looks like this:
The slice at height y has width x – 2 where (x,y) is a point on the graph of y = x4.
If y = x4 then x = y1/4, so the width of the slice is
x – 2 = y1/4 – 2.
The area of the slice is
(y1/4 – 2) Δ y.
The values that make sense are from y = 16 to y = 81.
Adding up the areas of all the slices and letting the number of slices approach infinity, the area of R is