Let *R* be the region bounded by the curve *y* = *x*^{4} and the lines *x* = 2 and *y* = 81. Write an integral expression for the area of *R*, using

(a) vertical slices

(b) horizontal slices

Answer

The region *R* looks like this:

The curve *y* = *x*^{4} 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 – *x*^{4}.

The area of this slice is

(81 – *x*^{4}) 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* = *x*^{4}.

If *y* = *x*^{4} then *x* = *y*^{1/4}, so the width of the slice is

*x* – 2 = *y*^{1/4} – 2.

The area of the slice is

(*y*^{1/4} – 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