Find the area bounded by the graphs of the following functions: f(x)=x^2, g(x)=x^3.

To find the area bounded by the graphs of the following
functions : f(x)=x^2, g(x)=x^3.

Let  us first find the
points where f(x) = y  = x^2  and g(x)= y = x^3
intersect.

At tthe inersection y coordinates are equal. So
x^2=x^3.

x^2-x^3 = 0. So
x^2(1-x).

Therefore x = 0 and 1-x = 0. Or x=
1.

So the point of intersections are at x= 0  and x=
1.

Thefore the area under f(x)  from x= 0 to x =1 is
calculated now:

The area A of the function f(x) between x =
a and x = b is given by:

A = F(b)-F(a), where F(x) =
Int(x)dx.

Int f(x) dx = Int x^2
dx.

F(x)  = 
(1/3)x^3.

Therefore  Area = F(b) -F(a)  =  (1/3) (1^3-0^3)
= 1/3.

Similarly area under g(x)  is given
by:

B = G(b)-G(a) , where G(x) = Int g(x)
dx.

G(x) = Int x^3 dx =
(1/4)x^4.

B = G(1)- G(0) = (1/4) (1^4-0^4) =
1/4.

Therefore the area enclosed betweeen f(x) and g(x) =
|A-B| = (1/3-1/4) = 1/12 sq units.