Use sine and cosine to parametrize the intersection of the surfaces x2 + y2 = 8 and z = 3x2.
Use x = 6 cos t and y = 6 sin t to parametrize the cylinder x2 + y2 = 8 in terms of t. Keep in mind that z can take on any value.
<cos t, sin t, z>
Parametrize the parabolic cylinder z = 3x2 in terms of x and y only. Notice here that x and y can take on any value.
<x, y, x2>
We wish to parametrize the intersection of these two curves in terms of sine and cosine. To do so, we set the parametrizations for each surface equal and solve for r(t) = <x(t), y(t), z(t)> in terms of cos t and sin t.
<6 cos t, 6 sin t, z> = <x, y, 3x2>
Solve for x(t), y(t), and z(t) in terms of cos t and sin t.
x(t) = cos t
y(t) = sin t
z(t) = (cos t)2
Write the vector parametrized intersection of the surfaces x2 + y2 = 8 and z = 3x2.
r(t) = < cos t, sin t, cos2t>, 0 ≤ t ≤ 2π