Find the work done by F = (x² + y)i + (y² + x)j + ze²k over the following paths from (2,0,0) to (2,0,4). a. The line segment x = 2, y = 0, 0≤z≤4 b.The helix r(t) = (2cos t)i + (2sin t)j + c. The x-axis from (2,0,0) to (0,0,0) followed by the parabola z = x², y = 0 from (0,0,0) to (2,0,4) b. Find The work done by F over the line segment is 3e4 +1. O A. B. C. df dt df dt df dt for F. = 1 //( 3 os ³t). + costsint+ 1 1/T 2t (2) ► = sint+ cost+ -e π k, 0≤t≤2 1 +7 ( sin ³t) + 2t 21/x - 21/x e π The work done by F over the helix is df 4t = cos ³t+2 costsint+ sin ³t+ 2e²1/7- dt 2t/x_2t/ e 2 + + df -= -8 cos ²t sint-4 sin ²t + 8 sin ²t cost + 4 cos dt 4t (2,0,4) 2t/x Z=X (2,0,0) (0,0,0)
Find the work done by F = (x² + y)i + (y² + x)j + ze²k over the following paths from (2,0,0) to (2,0,4). a. The line segment x = 2, y = 0, 0≤z≤4 b.The helix r(t) = (2cos t)i + (2sin t)j + c. The x-axis from (2,0,0) to (0,0,0) followed by the parabola z = x², y = 0 from (0,0,0) to (2,0,4) b. Find The work done by F over the line segment is 3e4 +1. O A. B. C. df dt df dt df dt for F. = 1 //( 3 os ³t). + costsint+ 1 1/T 2t (2) ► = sint+ cost+ -e π k, 0≤t≤2 1 +7 ( sin ³t) + 2t 21/x - 21/x e π The work done by F over the helix is df 4t = cos ³t+2 costsint+ sin ³t+ 2e²1/7- dt 2t/x_2t/ e 2 + + df -= -8 cos ²t sint-4 sin ²t + 8 sin ²t cost + 4 cos dt 4t (2,0,4) 2t/x Z=X (2,0,0) (0,0,0)
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Transcribed Image Text:Find the work done by F = (x² + y)i + (y² + x)j + ze²k over the
following paths from (2,0,0) to (2,0,4).
a. The line segment x = 2, y = 0, 0≤z≤4
b.The helix r(t) = (2cos t)i + (2sin t)j +
c.The x-axis from (2,0,0) to (0,0,0) followed by the parabola
z = x, y = 0 from (0,0,0) to (2,0,4)
b. Find
The work done by F over the line segment is 3e² + 1.
A.
B.
C.
df
dt
df
dt
df
dt
df
dt
df
D. dt
for F.
1
2t
1/2/(
cos ³t) + costsint+ (sin ³t) + ²t 12t/x_21/x
- e
π
sint + cost + e
π
= COS
π
1 1/T
-=-=-=e¹/x
= - 8 cos
k, 0≤t≤2n
³t+2 costsint+ sin ³t+
The work done by F over the helix is
4t
π²
2t/r_2t/r
-e
2tsint-4 sin ²t + 8 sin ²t cost + 4 cos²t+
(2,0,4)
4t 21/1
∙e
Z=X
Z
(2,0,0)
(0,0,0)
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