AUC06 Find the area bounded by y = 2 - (1/2)x2 and the x-axis.
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Graph the function. If you have the least bit of problem graphing this function go to previous sections concerned with graphing parabolas. This function is a parabola. It opens down and crosses the y-axis at y = 2. The limits on the integral have to be from where the curve crosses the x-axis on the negative side to where it crosses on the positive side. To find these points set and solve for x.
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Knowing how to graph this curve allows you to concentrate on the calculus part of the problem. If you have the least bit of problem graphing this function go to previous sections concerned with graphing parabolas.
The shaded area is the area desired so the integral is
Second Solution
There is a little faster, a little easier, and a
little less prone to error way of doing this problem. Remember the symmetry
that was so helpful in graphing parabolas? Not only is there a symmetry in the
graph of the curve between 0 and 2 and 0 and
,
but the area under the curve from 0 to 2 is the same as the area under the curve
from 0 to .
Therefore, the entire area between this curve and the x-axis is twice the
area calculation between x = 0 and x = 2. Notice the lack of
negative numbers in this solution.
In doing area problems look for symmetry that will make the problem easier and cut down on the amount of numbers you have to manipulate.
