1. **State the problem:** Find the part of the plane $z = x + 3$ that lies inside the cylinder defined by $x^2 + y^2 = 1$, and verify the point $P = (1, 0, 4)$ lies on this surface. 2. **Understand the surfaces:** - The plane is given by the equation $$z = x + 3$$. - The cylinder is given by $$x^2 + y^2 = 1$$, which is a circular cylinder of radius 1 centered on the $z$-axis. 3. **Find the region of the plane inside the cylinder:** - The cylinder restricts $x$ and $y$ such that $$x^2 + y^2 \leq 1$$. - On the plane, $z$ depends only on $x$, so the part of the plane inside the cylinder is all points $(x,y,z)$ where $$x^2 + y^2 \leq 1$$ and $$z = x + 3$$. 4. **Check the point $P = (1, 0, 4)$:** - Check if $P$ lies on the cylinder: $$1^2 + 0^2 = 1 \leq 1$$, so yes. - Check if $P$ lies on the plane: $$z = x + 3 = 1 + 3 = 4$$, which matches $z$ coordinate of $P$. - Therefore, $P$ lies on the part of the plane inside the cylinder. 5. **Summary:** - The part of the plane inside the cylinder is the set of points $$\{(x,y,z) \mid x^2 + y^2 \leq 1, z = x + 3\}$$. - The point $P = (1,0,4)$ lies on this surface. This describes the surface and confirms the point's location.