Calculus: Analytic Geometry and Calculus, with Vectors

276 Integrals

8 Using the fact that the pressure at depth d below the surface of water is
wd, but without using more formulas from the preceding problem, find the mag-
nitude of the force exerted upon one face of an isosceles right triangle submerged
in water so that one leg is horizontal and 5 feet below the surface while the other
leg extends 3 feet upward. fIns.: 18w.
9 According to an examination given at Cornell, the cost, in dollars per mile,
of improving the road from Alibab to the Babila oil field 400 miles down the road
is 10,000 plus 500 N/x, where x is the distance from Alibab. Find the total cost
of the improvement. flns.: 6 + millions.
10 A circle of radius a lies in the xy plane and has its center at the origin.
For each positive integer n, points Po, Pi, - -. ,P. are equally spaced on the
arc of the circle lying in the first quadrant and, for each k for which 1 5 k 5 n,
a vector rk is drawn from the origin to a point on the circle between Pk_i and Pk.
Show that

(1) lim

r, + r2 + ... + r,,=2a(i + A

n--> m n 7r

where, as usual, i and j are unit vectors on the x and y axes. Hint: Make use of

the fact that if

(2)

then

(3)

Ok =k7r, .,Ok = 2n,

rk=2a

(cos ON + sin 6k *j) DOk,

12 7r

where Ok lies between Ok_i and Ok. The left member of (1) is therefore the limit

of a Riemann sum.

11 One way to review Riemann integrals and make them seem simpler is to

learn about Riemann-Stieltjes integrals. Let f(t) and g(t) be defined over an

interval a S t < x, and let P be a partition of the interval a < t S x with parti-

tion points tk and intermediate points tk as in Section 4.2. If there is a number I

such that to each e > 0 there corresponds a a > 0 such that

n

(1) I Lf(tk)Ig(tk) - g()1 - I I <

kl

whenever JP1 < 5, then I is called the Riemann-Stieltjes integral off with respect

to g over the interval a 5 t < x and is denoted by

(2) Iaf(t) dg(t).

These integrals are very important in more advanced mathematics, and some

people think that they should at least be mentioned in elementary calculus.

Many people have devoted substantial parts of their lives to study of problems for

which f(t) = ts. Start picking up ideas by evaluating (2) when a = -1, x = 1,

f(t) = t2 + 2t + 3, and g(t) = sgn t. fins.: 6.

4.9 Simpson and other approximations to integrals When f is a

polynomial in x, and in some other cases, we can discover an elementary