Chapter 1. calc_tutorial_13_1_027

1.1 Problem Statement

{4,6,8}
{3,5,7}
{2,4,6}
$b*pow($c,2)
$c+$d

Use sine and cosine to parametrize the intersection of the surfaces x2 + y2 = $a and z = $bx2.

1.2 Step 1

Question Sequence

Question 1.1

Use x = $c cos t and y = $c sin t to parametrize the cylinder x2 + y2 = $a in terms of t. Keep in mind that z can take on any value.

<SFgqQUkJGdg=cos t, SFgqQUkJGdg=sin t, z>

Parametrize the parabolic cylinder z = $bx2 in terms of x and y only. Notice here that x and y can take on any value.

<x, y, iSba6t70dtA=x2>

Correct.
Incorrect.

1.3 Step 2

Question Sequence

Question 1.2

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.

<$c cos t, $c sin t, z> = <x, y, $bx2>

Solve for x(t), y(t), and z(t) in terms of cos t and sin t.

x(t) = SFgqQUkJGdg=cos t

y(t) = SFgqQUkJGdg=sin t

z(t) = U2GIbglD1oM=(cos t)2

Correct.
Incorrect.

1.4 Step 3

Question Sequence

Question 1.3

Write the vector parametrized intersection of the surfaces x2 + y2 = $a and z = $bx2.

r(t) = <SFgqQUkJGdg= cos t, SFgqQUkJGdg= sin t, U2GIbglD1oM= cos2t>, 0 ≤ t ≤ 2π

Correct.
Incorrect.