Horizontal curve is a curve viewed in the x and y plane, while a vertical curve is viewed in the y plane only, or viewed from the side. Think of it like a cake. the top is the horizontal and the front is the vertical

If the curve is on the xy-plane, finding an expression for dy/dx will give you the slope of a curve at a point.

A parabola is a U-shaped plane curve

theorems relating respectively to the volume and the surface area of a solid generated by the revolution of plane figure around an axis lying in the same plane. 1. when a plane figure is revolved about an axis lying in the same plane (but not cutting the area)The volume generated is equal to the product of the area and the length of the path described by the centre of gravity of the area . 2. When a plane curve is revolved about an axis lying in the same plane (but not cutting the curve) the surface area generated is equal to the product of the length of the curve and the length of the path of the centre of gravity of the curve.

A parabola refers to a symmetrical open plane curve that is formed by the intersection of the cone with a plane that is parallel to its side. The curve on the other hand refers to a line that gradually deviates from being straight for some or all of its length.