Calculus: Analytic Geometry and Calculus, with Vectors

4.6 Riemann-Cauchy integrals and work 257

Figure 4.692,which shows the rod before and after stretching, may be helpful.
Supposing that 0 < a < b, find the work done in stretching the rod from length
L + a to length L + b. .4ns.:

(bz- z).

TL_k

10 A conical container (see Figure 4.693) has height
a feet and base radius R feet. It is filled with sub-
stance (water or wheat, for example) which weighs w

L

' F

L+x

Figure 4.692 Figure 4.693

pounds per cubic foot and which must be elevated (by a pump or shovel or other

elevator) to a level H feet above the vertex. Suppose that H z a. Find the

work W required to accomplish the task. Hint: Start by making a partition of

the interval 0 < y _< a and calculating an approximation to the work required

to lift the material which constitutes a horizontal sheet or slab. All calculations

are based upon the fundamental idea that gravity pulls things downward, and that

the magnitude of the force on a thing is its weight. fns.:

W=w7rRza(H-a)3

4

Note that if V is the volume of the conical solid, then the answer can be put in

the form W = wV(H - *a).

11 Modify Problem 10 by replacing the conical container by a container

such that, for each y* for which 0 S y* S a, the plane having the equation y = y*

intersects the contents of the container in a set having area d(y*). Then set

up an integral for the work W. .4ns.:

W=w foa(H-y)A'(y)dy

12 In many problems involving motion of particles, we need the concept of

kinetic energy, or energy due to motion. This problem requires us to study and

learn a method by which we can use calculus to derive an important formula.

We suppose that, at time t = 0, a particle of mass in starts from rest, with kinetic

energy zero, at the origin of an x axis and is pulled in the direction of the positive

x axis by a force F of constant magnitude for which F = Ci at all times. We

suppose that no force other than F operates on the particle. Letting x denote

the coordinate of the particle at time t, we use the Newton law F = ma to obtain

the vector equation

z

(1) m dz i = ma = F = Ci.