Page 38 - Calc for the Clueless

Of course we do not have to approximate this integral, since it is easily done. However, we would need to
approximate, say, this one:

A theorem that is now mentioned much more often in calc I is the average value theorem. It says that if we have
an integrable function on the interval a < x < b, there exists a point c between a and b such that

We will demonstrate by picture.

Suppose we have the function as pictured. There is a point c where the two shaded areas are the same. Fill in the
top one in the bottom space. Thus the area of the rectangle equals the area under the curve. Area of the rectangle
is base times height. Base = b - a. Height = f(c).

Now divide by b - a.

That's it. Let's do an example.

Example 36—

2
Find the average value for f(x) = x , 2 < x < 5.

2
If I actually wanted to find the point c in the theorem that gives the average value, f(c) = (c = 13 = 3.6, which
1/2
clearly is between 2 and 5.
Let's do a word problem.

Example 37—

During the 12 hours of daylight, the temperature Fahrenheit is given by T = 60 + 4t - t /3. Find the average
2
temperature over the 12-hour period. The average temperature value