Questions: Find the vector equation that represents the curve of intersection of the cylinder (x^2+y^2=4) and the surface (z=xe^y). Write the equation so the (x(t)) term contains a (cos (t)) term. (x(t)=) (y(t)=) (z(t)=)

Find the vector equation that represents the curve of intersection of the cylinder (x^2+y^2=4) and the surface (z=xe^y).

Write the equation so the (x(t)) term contains a (cos (t)) term.

(x(t)=)
(y(t)=)
(z(t)=)

Transcript text: Find the vector equation that represents the curve of intersection of the cylinder $x^{2}+y^{2}=4$ and the surface $z=x e^{y}$. Write the equation so the $x(t)$ term contains a $\cos (t)$ term. \[ \begin{array}{l} x(t)=\square \\ y(t)=\square \\ z(t)=\square \end{array} \]

Solution

Solution Steps

Step 1: Parameterization of the Cylinder

To represent the curve of intersection of the cylinder $x^{2}+y^{2}=4$, we parameterize the circular base using trigonometric functions: \[ x(t) = 2 \cos(t) \] \[ y(t) = 2 \sin(t) \]

Step 2: Substitution into the Surface Equation

Next, we substitute the parameterized expressions for $x$ and $y$ into the equation of the surface $z = x e^{y}$: \[ z(t) = 2 \cos(t) e^{2 \sin(t)} \]

Final Answer

The vector equation that represents the curve of intersection is: \[ \begin{array}{l} x(t) = 2 \cos(t) \\ y(t) = 2 \sin(t) \\ z(t) = 2 \cos(t) e^{2 \sin(t)} \end{array} \] Thus, the final answer is: \[ \boxed{x(t) = 2 \cos(t), \; y(t) = 2 \sin(t), \; z(t) = 2 \cos(t) e^{2 \sin(t)}} \]

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