Expert: Socrates - 7/27/2006

Question
Find the area of the region bounded by the curve y=x(squared) and y=x

Answer
It is really a good idea to draw the graph of the curve
y = x^2 and the line y = x together. I can't draw the picture for you here, but if you draw the graphs, you will see that the line y=x lies above the curve y=x^2 between the points where they intersect. To find the points where they intersect , solve y=x^2 and y=x together. Substitute x in for y in the first equation and get x = x^2 , so
0 = x^2-x  and 0 = x(x-1). This gives x=0 and x=1 at the intersection points. Since the graph of y=x is above y=x^2 from x=0 to x=1, you integrate x - x^2 from 0 to 1. (always subtract the curve that lies underneath to find the area between)

S x - x^2 dx = (1/2)x^2 - (1/3)x^3

so we evaluate (1/2)x^2 - (1/3)x^3 at x = 1 and x = 0 , then take the difference.

At x=1 , (1/2) - (1/3) = 1/6

At x=0 , 0 - 0 = 0

The difference, 1/6 - 0 = 1/6 is the area
The answer is 1/6