Copyright

Use \cos(t) and \sin(t), with positive coefficients, to parametrize the intersection of the...

Question:

Use {eq}\cos(t) {/eq} and {eq}\sin(t) {/eq}, with positive coefficients, to parametrize the intersection of the surfaces {eq}x^2+y^2=4 {/eq} and {eq}z=6x^2 {/eq}.

Parametric Curves in Space


A parametric curve in space is given as a vector function of a single prameter {eq}\displaystyle \mathbf{r}(t)=\langle x(t), y(t), z(t) \rangle, t\in\mathbf{R}. {/eq}

To obtain the curve of intersection of two surfaces, we will solve the system of equations with the two equations of the surfaces.

If the surfaces involve a sum of squares of two variables we will involve sine and cosine functions in order to reduce the number of parameters to one.

Answer and Explanation:


To find the intersection between the circular cylinder {eq}\displaystyle x^2+y^2=4 {/eq} and the parabolic cylinder {eq}\displaystyle z=6x^2 {/eq}

we will solve the system of the two equation, {eq}\displaystyle x^2+y^2=4 \text{ and } z=6x^2. {/eq}

For this, we will use sine and cosine for x and y, because they appear as a sum of squares.

{eq}\displaystyle \boxed{ x=2\cos t, y=2\sin t, \text{ and } z=24\cos^2 t, 0\leq t\leq 2\pi}. {/eq}


Learn more about this topic:

Loading...
Evaluating Parametric Equations: Process & Examples

from Precalculus: High School

Chapter 24 / Lesson 3
3.9K

Related to this Question

Explore our homework questions and answers library