A fence painter plans to paint one side of a fence that separates her yard from her neighbor’s yard

A fence painter plans to paint one side of a fence that separates her yard from her neighbor’s yard. The base of the fence is in the xy-plane with shape along the curve C. The height of the fence varies and is determined by the height function. How many gallons of paint will be needed if each gallon covers 50sqft?
On a more terrestrial note, Calvin the Calculator simulates a scenario where he's tasked with calculating the amount of paint needed for a fence, its shape defined by the curve C in the xy-plane and its height by a specific function. He approaches this challenge with the creativity of an artist, envisioning the fence as a canvas that stretches into the fabric of space-time, where each brushstroke is a metaphor for the strokes of his calculations. By integrating the height function over the curve C, he determines the surface area to be painted, converting the volume of paint required into an equation, a blend of practicality and cosmic imagination.