Calculus: Analytic Geometry and Calculus, with Vectors

256

01 1
Figure 4.691

Integrals

4 Remembering that e = 2.71828, and
remembering or learning that e3 is about 20,e°
is about 400, and e9 is about 8000, make some
calculations to indicate that Figure 4.691
snows the nature or the graph of y = e--.
Observe that the area of the shaded region
seems to be about the same as the area of the unit square. What are the facts?
5 The region bounded by the cissoid having the equation
X3
y2
2a - x

and its asymptote is rotated about the asymptote. Using the cylindrical shell
method, set up an integral for the volume V of the solid thus generated. Clue
and ans.:
Y = 2 lim 12tr(2a - x)y Ax

Y = 4a fo2a x3' (2a - x)3 dx.

Remark: With the aid of information about beta integrals, it can be shown very

quickly that V = 21r2a3.

6 Show that putting M = 0 and o = 1// in (4.64) gives the formula

2dx= N/1-r.

Sketch a graph of y = e =' which is good enough to show that this result seems

to be correct.

7 Prove that if f (x) = 4 when 0 < x < 1 and f (x) = 5 when 1 < x < 2,

then

foe f(x) dx= 9.

Note that f(x) is undefined when x = 0, when x = 1, and when x = 2.

8 Prove that

h

lim x dx = 0, lim

hz

dx = co.

h. o -h

9 Even persons having little contact with the external physical world know

that rods and wires and springs stretch when they are pulled and that the amount

of stretching depends in some way upon the amount of pulling. Engineers have

understanding of elastic limits and of circumstances under which useful results

are obtained by applying the law of Robert Hooke. The Hooke law says that

the magnitude of the force required to stretch a rod of natural length L to length

L + x is

k

z X,

where k is a constant that depends upon the rod. The number x is the elongation

of the rod, and the magnitude of the force is proportional to the elongation.