Calculus AB and Calculus BC

CHAPTER 6 Definite Integrals

F. INTERPRETING ln x AS AN AREA

It is quite common to define ln x, the natural logarithm of x, as a definite integral, as follows:

Image

This integral can be interpreted as the area bounded above by the curve Image below by the t-axis, at the left by t = 1, and at the right by t = x (x > 1). See Figure N6–16.

Image

FIGURE N6–16

Note that if x = 1 the above definition yields ln 1 = 0, and if 0 < x < 1 we can rewrite as follows:

Image

showing that ln x < 0 if 0 < x < 1.

With this definition of ln x we can approximate ln x using rectangles or trapezoids.

EXAMPLE 35

Show that Image < ln 2 < 1.

SOLUTION: Using the definition of ln x above yields Image which we interpret as the area under Image above the t-axis, and bouned at the left by t = 1 and at the right by t = 2 (the shaded region in Figure N6–16). Since Image is strictly decreasing, the area of the inscribed rectangle (height Image width 1) is less than ln 2, which, in turn, is less than the area of the circumscribed rectangle (height 1, width 1). Thus

Image

EXAMPLE 36

Find L(5), R(5), and T(5) for Image

SOLUTION: Noting that for n = 5 subintervals on the interval [1,6] we have Δx = 1, we make a table of values for Image

x

1

2

3

4

5

6

f (x)

120

60

40

30

24

20

Then:

Image

NOTE: The calculator finds that Image is approximately 215.011.