Parameterization
Parameterize the unit circle (radius 1) centered at the origin, traced counterclockwise.
x = cos(t), y = sin(t), 0 <= t <= 2pi
For the parametrization C: x = 3t, y = 4t, 0 <= t <= 1, find ds.
ds = sqrt(3^2 + 4^2) dt = 5 dt
When integrating over a piecewise smooth curve C = C1 + C2, how do you compute the total line integral?
Partition it: compute the integral over each smooth piece separately and add the results. Integral over C = Integral over C1 + Integral over C2.
Find the parametric equations for a circle of radius 3 centered at the origin, traced counterclockwise.
x = 3cos(t), y = 3sin(t), 0 <= t <= 2pi
Evaluate the integral of (x + y) ds, where C is the line segment from (0,0) to (3,4).
17.5 (Hint: parameterize as x=3t, y=4t; ds=5dt; integral of 35t dt from 0 to 1)
For path C1 from (0,0) to (1,2) parameterized as x = t, y = 2t, 0 <= t <= 1, what is ds?
ds = sqrt(5) dt
Parameterize the line segment from (0,0) to (3,4).
x = 3t, y = 4t, 0 <= t <= 1
Evaluate the integral of x ds, where C is the unit circle x = cos(t), y = sin(t), 0 <= t <= 2pi.
0 (The integral of cos(t) * 1 dt from 0 to 2pi = 0 by symmetry)
Evaluate the integral of x^2 ds over C1, where C1 is the line segment from (0,0) to (1,2), parameterized as x = t, y = 2t, 0 <= t <= 1.
sqrt(5)/3 (Integral of t^2 * sqrt(5) dt from 0 to 1 = sqrt(5) * [t^3/3] from 0 to 1 = sqrt(5)/3)
Parameterize the vertical line segment from (1,2) to (1,4).
x = 1, y = t, 2 <= t <= 4
For C: x = t, y = 2t, 0 <= t <= 1, find ds.
ds = sqrt(1^2 + 2^2) dt = sqrt(5) dt
Evaluate the integral of x^2 ds over C2, the vertical segment from (1,2) to (1,4), parameterized as x = 1, y = t, 2 <= t <= 4.
2 (x=1, so x^2=1; ds=dt; integral of 1 dt from 2 to 4 = [t] from 2 to 4 = 2)
Parameterize the line segment from (0,0) to (1,2).
x = t, y = 2t, 0 <= t <= 1
Evaluate the integral of (x^2 + y^2) ds where C is the unit circle x = cos(t), y = sin(t), 0 <= t <= 2pi.
2pi (Since x^2 + y^2 = 1 on the unit circle and ds = dt, integral = integral from 0 to 2pi of 1 dt = 2pi)
Evaluate the integral of x^2 ds where C consists of the line segment from (0,0) to (1,2) followed by the vertical segment from (1,2) to (1,4).
sqrt(5)/3 + 2 (Partition: integral over C1 = sqrt(5)/3, integral over C2 = 2; Total = sqrt(5)/3 + 2 approximately 2.745)