calc_tutorial_13_1_027

 
Problem Statement

{4,6,8}
{3,5,7}
{2,4,6}
3*pow(6,2)
6+108

Use sine and cosine to parametrize the intersection of the surfaces x2 + y2 = 6 and z = 3x2.

 
Step 1

Question Sequence

Question 1

Use x = 6 cos t and y = 6 sin t to parametrize the cylinder x2 + y2 = 6 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>

Correct.
Incorrect.

 
Step 2

Question Sequence

Question 2

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

Correct.
Incorrect.

 
Step 3

Question Sequence

Question 3

Write the vector parametrized intersection of the surfaces x2 + y2 = 6 and z = 3x2.

r(t) = < cos t, sin t, cos2t>, 0 ≤ t ≤ 2π

Correct.
Incorrect.