In each part, evaluate the integral ∫ C y d x + z d y − x d z along the stated curve (a) The line segment from (0, 0, 0) to (1, 1, 1). (b) The twisted cubic x = t , y = t 2 , z = t 3 from (0, 0, 0) to (1, 1, 1). (c) the helix x = cos π t , y = sin π t , z = t from (1, 0, 0) to ( − 1 , 0 , 1 ) .
In each part, evaluate the integral ∫ C y d x + z d y − x d z along the stated curve (a) The line segment from (0, 0, 0) to (1, 1, 1). (b) The twisted cubic x = t , y = t 2 , z = t 3 from (0, 0, 0) to (1, 1, 1). (c) the helix x = cos π t , y = sin π t , z = t from (1, 0, 0) to ( − 1 , 0 , 1 ) .
In each part, evaluate the integral
∫
C
y
d
x
+
z
d
y
−
x
d
z
along the stated curve
(a) The line segment from (0, 0, 0) to (1, 1, 1).
(b) The twisted cubic
x
=
t
,
y
=
t
2
,
z
=
t
3
from (0, 0, 0) to (1, 1, 1).
(c) the helix
x
=
cos
π
t
,
y
=
sin
π
t
,
z
=
t
from (1,
0,
0) to (
−
1
,
0
,
1
)
.
With differentiation, one of the major concepts of calculus. Integration involves the calculation of an integral, which is useful to find many quantities such as areas, volumes, and displacement.
Determine the intersection point of the lines tangent to the curve r (t) = (senπt, 2senπt, cosπt)at points t = 0 and t = 0,5.
Find the tangential and normal components.
r(t)=(t+1)i+2tj+t^ 2 × k , t = 1
r(t) = (t cos t) i + (t sin t) j + t^2 × k , t = 0
r(t)=t^2 i+(t+(1/3)t^ 3 )j+(t-(1/3)t^ 3 )k , t = 0
r(t)=(e^ t cos t) i +(e^ t sin t)j+ √2e^t k , t=0
Express dw/ðu and əw/dv as functions of u and v both by using the Chain Rule and expressing w directly in terms of u
and v before differentiating.
w = In(x? + y? + z²), x= e" sinu, y=e" cos u, z= ue ": (u,v) = (-2,0)
Need a deep-dive on the concept behind this application? Look no further. Learn more about this topic, calculus and related others by exploring similar questions and additional content below.
Numerical Integration Introduction l Trapezoidal Rule Simpson's 1/3 Rule l Simpson's 3/8 l GATE 2021; Author: GATE Lectures by Dishank;https://www.youtube.com/watch?v=zadUB3NwFtQ;License: Standard YouTube License, CC-BY