Question

Sketch the space curve represented by the intersection of the surfaces. Then represent the curve by a vector-valued function using the given parameter. $$\text { Surfaces } \quad \text { Parameter }$$ $$ x^{2}+y^{2}=4, \quad z=x^{2} \quad x=2 \sin t $$

Short answer

The vector-valued function for the space curve is \(r(t) = \langle 2\sin t, 2\cos t, 4\sin^{2} t \rangle\).

Step-by-step solution

Analyze the First Surface

The equation \(x^{2} + y^{2} = 4\) represents a cylinder in 3D space with a radius of 2. This can be visualized as a circle centered at the origin in the xy-plane, which is then extruded along the z-axis to form a cylinder.

Analyze the Second Surface

The equation \(z = x^{2}\) is a parabolic surface, which opens upward along the z-axis. For every value of \(x\), \(z\) is the square of \(x\), introducing a curvature into the surface.

Intersection of the Surfaces

The space curve represented by the intersection of the two surfaces is a curved line that weaves up and down through the interior of the cylinder. When the surfaces \(x^{2} + y^{2} = 4\) and \(z = x^{2}\) intersect, we get points \((x, y, z)\) that satisfy both equations.

Use of the Parameter

We use the provided parameter \(x = 2\sin t\) to further simplify the equations. Substituting this into our first equation, we get \(y^{2} = 4 - x^{2} = 4 - 4\sin^{2} t = 4\cos^{2} t\). Taking the positive root (since \(y\) lies in the first and fourth quadrant), we get \(y = 2\cos t\). Substituting \(x = 2\sin t\) into the second equation, we get \(z = x^{2} = 4\sin^{2} t\). The vector-valued function representing the curve is therefore \(r(t) = \langle 2\sin t, 2\cos t, 4\sin^{2} t \rangle\).

Sketch the Space Curve

This final expression forms an ellipse around the circle in the xy-plane, ranging between a maximum and minimum height in the z direction. To sketch the curve, start by drawing the cylinder and the parabolic surface based on their individual equations. Then, inside the cylinder, draw a spiral space curve that oscillates along the z-axis.