THE NATURAL LOGARITHM
191
23.54
Find the length of the curve y = jjc
2 - Jin x between x = 1 and x = 8.
So, the arc length is
iln2.
23.55
Graph y = x
2 - 18 In x.
23.56
Show that the area under y = 1 Ix from x = a to x = b is the same as the area under that curve from
x — ka to x = kb for any k > 0.
Fig. 23-7
On the other hand,
In
In.
In
In
In
In
23.57
Use the Trapezoidal Rule, with n = 10, to approximate In 2 =
For
The Trapezoidal Rule yields
(The actual value, correct to four decimal places, is 0.6931.)
23.58
If y =
find y'.
Use logarithmic differentiation. In y = In x + 2 In (1 - *
2 ) - 5 In (1 + x
2 ).
In
Hence,
So,
23.59
Find the volume of the solid generated when the region under y = 1 /x
2 , above the *-axis, between x = 1
and x = 2, is rotated around the y-axis.
By the cylindrical shell formula, V= 2ir I? xy dx = 2n
See Fig. 23-7. y' = 2x - 18/x, y" = 2 + 18/x
2 . Setting y' = Q, we find that x
2 = 9, x = 3, y = 918 In 3=-10. Hence, by the second-derivative test, there is a minimum at A: = 3. As x-*+<*>, y =
As x—»0
+ , In*—»— 0° and y—*+x.
= 0.0750 + (0.0909 + 0.0833 + • • • + 0.0526) = 0.0750 + 0.6187 = 0.6937.
dx = 2ir In x = 27r(ln2-0) = 2irln2.
dx = In x ]** = In kb - In ka =