Surface integrals are used for a wide variety of applications, particularly in physics, but an often overlooked application is determining the surface area of various shapes. At some point in our education, we are presented with formulas for the surface area of common shapes like spheres, cylinders, cones, and rectangular prisms (boxes). PIck one of these and use a surface integral to verify the known formula for its surface area. Feel free to look up the surface area formulas online or in a book; I don't expect you to know them offhand! Surface area of the cylinder and start with the center of the cylinder to be the origin. And the equation of the cylinder is 2 r h + 2 r.