What is the area enclosed by the functions x^2 and sqrt(x)?
First, let's see how the plot of the functions looks like (draw on whiteboard). Next, let's calculate where the functions intersect by setting x2 = sqrt(x) and solving for x (manipulate by squaring both sides and get x4=x and combine to form x(x3-1)=0 which gives x=0 or 1). Finally, find the area by integrating the difference of the functions between these two points (integral from 0 to 1 of sqrt(x)-x2 dx = [2/3 x3/2 -1/3 x3] evaluated from 0 to 1 = 2/3-1/3 = 1/3). Therefore, the area enclosed by the functions x^2 and sqrt(x) is 1/3.
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