We need to find the area bounded between the parabola y equals x squared and the line y equals x. Let's start by plotting both functions and identifying their intersection points.
To find the intersection points, we set x squared equal to x. This gives us x squared minus x equals zero, which factors as x times x minus one equals zero. Therefore, x equals zero and x equals one are our intersection points at coordinates zero comma zero and one comma one.
Now we need to identify which function is on top in the interval from zero to one. Let's test at x equals zero point five. For the line y equals x, we get zero point five. For the parabola y equals x squared, we get zero point two five. Since zero point five is greater than zero point two five, the line is above the parabola in this interval.
To find the area between the curves, we use the integral formula: area equals the integral from a to b of upper function minus lower function dx. Substituting our functions, we get the integral from zero to one of x minus x squared dx. Evaluating this integral gives us x squared over two minus x cubed over three, evaluated from zero to one. This equals one half minus one third, which simplifies to one sixth.
To summarize: we found the intersection points of the two curves at zero comma zero and one comma one. We determined that y equals x is the upper curve in this interval. We set up the definite integral from zero to one of x minus x squared dx, and evaluated it to find that the area between the curves is one sixth.