Sage Worksheet: Problem13.2.30

var('x y z t')

(x, y, z, t)

(x, y, z, t)

mass = integral( (t^2 + cos(t)^2 + sin(t)^2) *sqrt(2) , 0, 2*pi)
mass

32(3+43)2

32(3+43)2

mass = integral( (t^2 + 1) *sqrt(2) , 0, 2*pi)
mass

32(3+43)2

32(3+43)2

n(mass)

125817757876244

125817757876244

xbar = 1/mass * integral( t * (t^2 + 1) *sqrt(2) , 0, 2*pi)
ybar = 1/mass * integral( cos(t) * (t^2 + 1) *sqrt(2) , 0, 2*pi)
zbar = 1/mass * integral( sin(t) * (t^2 + 1) *sqrt(2) , 0, 2*pi)
(xbar, ybar, zbar)

3(24+2)(3+43)6(3+43)−62(3+43)

3(24+2)(3+43)6(3+43)−62(3+43)

factor(xbar),factor(ybar),factor(zbar)

3(42+3)(22+1)61(42+3)−6(42+3)

3(42+3)(22+1)61(42+3)−6(42+3)

(n(xbar), n(ybar), n(zbar))

4601452903992300141248199400547−0443744305569546

4601452903992300141248199400547−0443744305569546

density(x,y,z) = x^2 + y^2 + z^2

r = (t, cos(t), sin(t))

d = [diff(a,t) for a in r] # differentiate the three components of r with respect to t
d

[1, -sin(t), cos(t)]

[1, -sin(t), cos(t)]

s=sqrt(sum([c^2 for c in d])) # sqrt of sum of squares of those derivatives
s

sqrt(sin(t)^2 + cos(t)^2 + 1)

sqrt(sin(t)^2 + cos(t)^2 + 1)

integral(density(r[0],r[1],r[2])*s,0,2*pi)

32(3+43)2

32(3+43)2

factor(integral(density(r[0],r[1],r[2])*s,0,2*pi))

32(42+3)2

32(42+3)2

integral( (t^2 + sin(t)^2 + cos(t)^2) *sqrt(2) , 0, 2*pi)

2/3*(3*pi + 4*pi^3)*sqrt(2)

2/3*(3*pi + 4*pi^3)*sqrt(2)