Solved problems on line integrals

Q1)
F(x, y) = (exp(y), xexp(y) – 2ysin(y2)
r(t) = (t3, cos(πt)) where -1 ≤ t ≤1
x = t3, y = cos(πt)
r’(t) = 3t2, -πsin(πt)
F(r(t)) = (exp(y), xexp(y) – 2ysin(y2) * (3t2, -πsin(πt))
= ∫C
❑
F (x , y )∗dr = ∫−1
1
¿¿exp(y), xexp(y) – 2ysin(y2) * (3t2, -πsin(πt))dt
= ∫−1
1
3 t2 ¿¿
= ∫−1
1
3 t2 exp(cos (πt ))−π sin ⁡(πt )t3 exp (cos (πt )+2 π sin ⁡(πt )cos (πt )sin (cos (πt ))2
]dt
= [ cos (2 πt )+1
2 +t 3 exp ¿¿]-11
= 1.03678 – 0.6322
= 0.40458
Q2)
F(x, y, z) = (z-x, -y, z)
r(t) = (t, t, sin(t))
x = t, y = t, z = sin(t)
r’(t) = 1, 1, cos(t)
F(r(t)) = sin(t)-t, -t, sin(t)
= ∫γ
❑
F (x , y , z )∗dr = ∫0
1
¿¿sin(t)-t, -t, sin(t)]*[1, 1, cos(t)]dt
= ∫0
1
¿¿ 1(sin(t)-t)+1* -t +cos(t)* sin(t)]dt

= ∫0
1
¿¿ (sin(t) – 2t +cos(t)sin(t)]dt
= [ sin2 (t )−2(cos (t )+t2 )
2 +C]01
= 0.000304586 – 2*1.99985
= -1.9997
≈−2
Q3)
F(x, y)= [cos)x)-y(x2), xy2,ln(y+1)]
r(t) = (cos(t), sin(t))
X = cos(t), y = sin(t)
r’(t) = [-sin(t), cos(t)]
F(r(t)) = cos(cos(t)) – sin(t)(cos(t))2, cos(t)(sin(t))2 + ln(sin(t) + 1)
= ∫γ
❑
F (x , y )∗dr = ∫0
1
¿¿ cos(cos(t)) – sin(t)(cos(t))2, cos(t)(sin(t))2 + ln(sin(t) + 1)]* [-sin(t),
cos(t)]dt
= ∫0
1
¿¿
[(sin(t)+1)ln(sin(t)+1)+sin(cos(t))- sin(4t)/16 – sin(t)+t/4+C]0π
= -0.034905 + π
4
Q4)
a) U(x, y, z) = (xyz – x2, y2 + z2, 2xz)
∇u=(∂
∂ x , ∂
∂ y , ∂
∂ z )∗(xyz – x 2, y 2+z 2 , 2 xz )
∂
∂ x (xyz−x2 )+∂
∂ y (y2+z2 )+∂
∂ z (2 xz)
yz−2x+2y+2x
¿yz +2 y
if r =(x , y , z )∧r=¿r ∨¿