Example 4: Continuous Accumulation

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The Trystero Trombone Society is holding a fund-raiser. Outside of their modest offices, they have constructed a large billboard showing the outline of a trombone. As pledges gradually accumulate, the members hope to fill in the trombone with a representative amount of red paint. The idea is that the painted area inside the trombone will display the total number of dollars that have been contributed.

The function  y = 1/x  has been chosen to reproduce the trombone's elegant curves. The resulting billboard is shown below.

Eager to get the fund-raiser rolling, Edie Maas, the Society's president, slaps  $2  from her own pocket onto the table. Full of excitement, the members of the Society hurry out to the billboard with paint brushes in hand. But Edie, exercising the cool head that has won her the Society's executive position, pauses the group before the sign. She wonders: How much of the trombone should be painted to accurately display her  $2  contribution?

She decides, first of all, that they must start at the point marked  x = 1, and then paint to the right to some  x = a , since the trombone flares widely to the left of  x = 1 , up and out of reach. The problem, then, is to determine the value of  a  which will give the painted region an area of exactly two. Edie scratches her head, and sends everyone unhappily back inside.

Edie knows how to find the area of a rectangle: base x height. She even remembers how to find the area of a trapezoid: (average of the two bases) x height. But the region they wish to paint has a curving top and bottom, quite unlike a rectangle or a trapezoid. Concerned about the silence that has overtaken the otherwise boisterous group, Edie decides to estimate, using the figure below.

In the figure, a trapezoid with vertices at  (1, 1) , (1, -1) , (a, -1/a) , and  (a, 1/a)  has been drawn in. The trapezoid looks like it has an area close to the shaded region – just a little bigger. So Edie computes this trapezoid's area (the height being sideways here):

A	= [average of the two bases](height)
	= [(1/2)((1 – (-1)) + (1/a – (-1/a)))](a – 1)
	= [(1/2)(2 + 2/a)](a – 1)
	= [1 + 1/a](a – 1)
	= a – 1/a.

Setting  a – 1/a = 2 , the desired area, she arrives at a quadratic equation for the approximate value of  a: a² – 2a – 1 = 0 . Solving this with the quadratic formula, she finds one root to the right of  x = 1 :

Since the trapezoid she drew in was a little too big, Edie knows that this approximate value is a little too small. The approximation also gets worse as  a  gets bigger. (Do you see why?) Fearing reprisals from the precision marching cell of the Society, Edie quickly looks for a way to improve her estimate.

Something a little larger than  2.414... . Could it possibly be  2.718... , the number  e  she heard a table of mathematicians joking about so loudly the last time she was at the coffee shop? Glancing over her shoulder at the Society's members, many of whom are now staring dejectedly at the floor, Edie decides to do a quick check.

Does  a = e  give an area of two? With a definite value to use for  a, Edie can get a much more accurate approximation of the area inside of the trombone. She sketches the following figure:

The area is now approximated with ten smaller trapezoids, each with sides much closer to the curve than the one used before. This should give a good estimate of the area.

Edie sees that the (sideways) height of each of the trapezoids will be the same:  (e – 1)/10 . The (vertical) bases of the trapezoids will cross the axis, from left to right, at  x = 1 ,  x = 1 + (e – 1)/10 ,  x = 1 + 2(e – 1)/10 , ... , and  x = e . The corresponding lengths of the bases, computed using  y = 1/x  and then doubling to give the lengths above and below the axis, will be:

2(1/1) = 2 2(1/(1+ (e – 1)/10) = 2/(1 + (e – 1)/10) 2(1/1 + 2(e – 1)/10) = 2/(1 + 2(e – 1)/10) ... 2(1/e) = 2/e .

Edie finds the areas of each of the ten trapezoids and adds them together. (Try it!) The result, rounding to three decimal places, is: 2.004 . If the correct value of  a  is not  e , then it must be something very close!

Edie calls the Society to order and announces in an authoritative tone that they will paint the inside of the trombone from  x = 1  to  x = e . The members of the Society jump to their feet and, amid a fanfare of trombones, march in joyous formation out to the sign.

It is not only the number  e, but also the natural exponential function  e^x  that turns up in surprising places. The next example points out a way in which the celebrated exponential may be observed in your daily affairs.