John is stressed out. He's got all kinds of projects going on, and they're all falling due around the same time. He's trying to stay calm, but despite all of his yoga techniques, relaxation methods, and sudden switch to decaffeinated coffee, his nerves are still on edge. Forget the 50-page paper due in two weeks and the 90-minute oral presentation next Monday morning; he still hasn't done laundry in weeks, and his stink is beginning to turn heads. Because of all the stress (and possibly due to his lack of bathing), John's started to lose his hair.

Below is a chart representing John's rate of hair loss (in follicles per day) on various days throughout a two-week period. Use 6 trapezoids to approximate John's total hair loss over that traumatic 14-day period.

       

Difficulty:

Solution

You may be tempted to use the Trapezoid Rule, but you can't use that handy formula, because not all of the trapezoids have the same width. Between day 1 and 4, for example, the width of the approximating trapezoid will be 3, but the next trapezoid will be 6 – 4 = 2 units wide. Therefore, you need to calculate each trapezoid's area separately, knowing that the area of a trapzoid is equal to one-half of the product of the width of the trapezoid and the sum of the bases:

Remember, this is only an estimate--there's a lot we don't know about ol' John's hair loss.