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.
Q 2.
Let C₁ be the straight line from the point (1,0) to the point (0, 1) in Figure 1. Let C₂ be an
oriented and closed path in Figure 1.
(a)
(b)
Evaluate the line integral of F = 4xi + 2xj along C₁.
Evaluate the line integral of F = sin(2x)i + ej along C₂.
Figure 1: A closed and oriented path
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
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