find the area the region bounded by the graphs of the given equation y=81x,y=x^5,x=0,x=3

Answer 1

The area is between the function y=x^5 and y=81x, between the limits x=0 and x=3.

The functions are all in the first quadrant (no values negative), and the two curves y=81x and y=x^5 intersect at x=3, the upper integration limit, which is a "lucky coincidence".

The area function a(x), is therefore
a(x)=81x-x^5
You would integrate a(x) from 0 to 3 to find the area, thus
Area = ∫a(x)dx [x=0,3]
=∫(81x-x^5)dx [x=0,3]

Here's an image to give you an idea of the area:
http://img11.imageshack.us/img11/6060/1290966183.png