# Calculating The Surface Area of a Smooth Object Defined by a Function Using Calculus $ A = \int \limits_a^b 2\pi y \sqrt {f'(x)^2+1} \, dx $ This follows the formula for [[Determining the Length of a Section of a Curve]]. Individual curve sections are multiplied by $2\pi y$ to calculate the surface area they form when revolved around the $x$ axis. Note that this method excludes the object's leftmost and rightmost sides (if revolving a line around the $x$ axis you'll get a cone, and this formula does not include the surface area of the cone's top and bottom).