Find the area between the curves C_1, C_2 and the lines x=0 and x=1, where C_1 is the curve y = x^2 and C_2 is the curve y = x^3.

We start by drawing a diagram which illustrates the question. First draw the x-y plane and the two curves curve y = x^2 and y = x^3. Notice that the two curves intersect at x=0 and x=1, and in the range 0<x<1, the curve y = x^2 is above the curve y = x^3. Shade in the area which we are asked to find.
In order to find the area under y=x^2, call it A_1, we integrate the function y^2 between the limits of x = 0 and x = 1. [Explain the integration procedure if student is unsure of how it works.] This gives the answer A_1 = 1/3. Now, we integrate the function y = x^3 between limits x=0 and x = 1 to find the area A_2 under the second curve. The result is A_2 = 1/4.
Looking back to our diagram, it is clear that the area we are after is the difference between the two areas we have calculated, i.e. A_1 - A_2 = 1/12.