Calculus: Analytic Geometry and Calculus, with Vectors

282 Integrals

Remark: The last cases show results obtained from an electronic computer that
makes 8D calculations. When n is large, rounding errors seriously affect the
last digit or digits kept.
S A loaded freighter is anchored in still water. At water level, the boat is
200 feet long and, for each k = 0, 1, 2, ,20, has breadth yk at distance 10k
feet from the prow. Assign semireasonable numerical values to the numbers
yk and do not allow anyone toclaim that you have not partially designed a boat.
Then use Mr. Simpson's idea to approximate the area of the water-level section
of your boat. Finally, recall an exploit of Archimedes and make an estimate of
the number of tons of freight that should be removed in order to raise your boat
1 foot.
6 Use the Simpson formula to obtain decimal approximations to the follow-
ing integrals. Keep two and three decimal places in the calculations, use a
slide rule if possible, and use the value of it given in parentheses.

(a)

J04

x2 dx, (n = 4) (b)

fo4

x3 dx, (n = 4)

loll

(c) fioo xdx, (n = 2)

x

(d) fog sin x dx, (n = 6)

sin x

(e) f l/sin x dx, (n = 6) (f)

f

dx, (n = 6)

o x

(g) fof (1 + x2)35 dx, (n = 4) (h) 101 e -l' dx, (n = 10)

(i) fi e-' dx, (n = 10) f 23 e x= dx, (n= 4)

7 Using the fact that x22 > 3x when x > 3, show that

e='dx<

e3zdx=-le_3z -le9_1^1 =^1

13' L^133 - 3 (20)3 24,000

8 Using the notation and ideas employed to derive (4.953), prove that if the

graph of the function f for which

(1) f(x) = K(x - x1)3 + 4(x - x,)2 + B(x - x1) + C

contains the three points Po(xo, yo), P1(x1, YI), P2(x2, y2), then

(2) f(x) dx =

h[yo+4y1+y2]

f 72 3

Remark: This result shows that the error term is zero and the Simpson formula

gives the exact value of the integral when f is a polynomial of degree three or less.

Thus we catch the idea that the Simpson formula gives good approximations even

when the integrand cannot be closely approximated over the intervals xk S x <

Xk+2 by quadratic polynomials but can be closely approximated over the intervals

by cubic polynomials. Further investigation shows that if we add to the right

member of (1) an integrable term ¢(x) for which 14(x)l 5 M(x - x1)4, then (2)

will contain an error term a for which lei -< (3)Mhb.