Answer
Their intersection is the given parametric curve, which wraps around the cylinder and follows the parabola \(z=x^2\).
\[
z=x^2 \quad Blue\\
x^2+y^2=1 \quad Red
\]
Work Step by Step
We have:
\( x=\sin t, \quad y=\cos t, \quad z=\sin^2 t \),
and:
\( z=x^2, \quad x^2+y^2=1 \).
Lets start by manipulating \(x\) and \(y\) to verify \(x^2+y^2=1\):
\[
x^2+y^2=(\sin t)^2+(\cos t)^2=\sin^2t+\cos^2t
\]
Since \(\sin^2t+\cos^2t=1\), we have shown that:
\(x^2+y^2=1\)
Now, let's manipulate \(x\) to verify \(z=x^2\):
\[
x=\sin t \implies x^2=(\sin t)^2\]
Since \(z=\sin^2t\), it follows that:
\[ z=x^2\]
