Thomas Calculus 13th [Solutions]

462 Chapter 6 Applications of Definite Integrals
37-42. Example CAS commands:
Maple:
with( plots );
with( Student[Calculus1] );
with( student );
f : x - sqrt(1-x^2);a : -1;
b : 1;
N : [2, 4, 8];
for n in N do
xx : [seq( a i*(b-a)/n, i 0..n )];
pts : [seq([x,f (x)], x xx)];
L : simplify(add( distance(pts[i 1],pts[i]), i 1..n ));
# (b)
T : sprintf("#37(a) (Section 6.3)\nn %3d L %8.5f \n", n, L );
P[n] : plot( [f (x),pts], x a..b, title T ): # (a)
end do:
display( [seq(P[n], n N)], insequence true, scaling constrained );
L : ArcLength( f(x), x a..b, output integral ):
L evalf ( L ); # (c)
Mathematica: (assigned function and values for a, b, and n may vary)
Clear[x, f ]
{a, b} { 1, 1}; f[x _] Sqrt[1
2
x ]
p1 Plot[f[x], {x, a, b}]
n 8;
pts Table[{xn, f[xn]}, {xn, a, b, (b a)/n}]/ / N
Show[p1,Graphics[{Line[pts]}]}]
Sum[ Sqrt[ (pts[[i 1, 1]]
2
pts[[i, 1]]) (pts[[i 1, 2]]
2
pts[[i, 2]]) ], {i, 1, n}]
NIntegrate[Sqrt[ 1
2
f '[x] ], {x, a, b}]
6.4 AREAS OF SURFACES OF REVOLUTION
1. (a)
dy
dx
2
2 dy 4
sec x sec x
dx
(b)
/4 4
S 2 (tan x) 1 sec x dx
0
(c) S 3.84
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