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CHAPTER 29
29.42
Assume that an object A, starting at (a, 0), is connected by a string of constant length a to an object B, starting at
(0,0). As B moves up the _y-axis, A traces out a curve called a tractrix. Find its equation.
Let A be at (x, y). The string must be tangent to the curve. From Fig. 29-11, the slope of the tangent line is
So, by Problem 29.41,
Since y = 0 when x = a, C = 0
Fig. 29-11
In a disk of radius a, a chord b units from the center cuts off a region of the disk called a segment. Find a formula
for the area of the segment.
From a diagram, the area
Then,
By Problem 29.1, this is
29.43
29.44
29.45
The region under y = \l(x
2 + l), above the x-axis, between x = 0 and x = l, is revolved about the
x-axis. Find the volume of the resulting solid.
By the disk formula
Let
Then
Then
Find
Let
Thus,
[More generally, the above change of variable, z = tan (x/2), converts the indefinite integral of any rational
function of sin x and cos x into the indefinite integral of a rational function of z.]
In
dx = J a cos 0 • a cos 9 d0 = a J cos 6 d0.
Let jc = a sin 0, dx = a cos 0 dfl.
Hence,
* = tan 0, dx = sec
2 6 d6.
x = 2tan * z,