16) Calcule Rαf ds si fyαson los que indican
(a) f(x, y, z) = x+y+z , α(t) =
(2t, 0,0) si 06t61
(2,3t−3,0) si 16t62
(2,3,2t−4) si 26t63
(b) f(x, y, z)=2x−√y+ 2z2, α(t) = (t, t2,0) si 06t61
(1,1, t −1) si 16t63
Sol
(a) f(x, y, z) = x+y+z , α(t) =
(2t, 0,0), t [0,1]
(2,3t−3,0), t [1,2]
(2,3,2t−4), t [2,3]
i)
I1=α(t) = (2t, 0,0)
α0(t) = (2,0,0) ⇒Z1
0
2t(2) dt =4t2
2|1
0= 2
kα0(t)k= 2
f(α(t)) = 2t
ii)
I2=α(t) = (2,3t−3,0)
α0(t) = (0,3,0) ⇒Z2
1
(3t−1)(3) dt =9t2
2−3t|2
1
kα0(t)k= 3 = 9(4)
2−6−9
2−3
= (12) −3
2=21
2
f(α(t)) = 3t−1
iii)
I3=α(t) = (2,3,2t−4)
α0(t) = (0,0,2) ⇒Z3
2
(2t+ 1)(2) dt =4t2
2+ 2t|3
2
kα0(t)k= 2 = 4(9)
2+ 2(3)−4(4)
2+ 4
= (24) −(12) = 12
f(α(t)) = 2t+ 1
∴I1+I2+I3= 2 + 21
2+ 12 = 49
2
1
