Surface Area of Revolution | Rotating Curves Around the X and Y Axis

In this lesson, we walk through how to find the surface area of revolution, which is the surface formed when a curve is rotated around an axis. Unlike finding the volume of a solid of revolution, we are focusing only on the outer surface created by the points on the curve as they spin around the axis of rotation. We start by building an understanding of where the surface area formula comes from, connecting it to the arc length formula you already know. The key idea is that each cross-section of the surface is the circumference of a circle, and the radius of that circle depends on which axis you are rotating around. From there, we work through four fully solved examples that cover the most common situations you will encounter. The examples include rotating about the x-axis using a function of x, rotating about the x-axis using a function of y, rotating about the y-axis using a function of y, rotating about the y-axis using a function of x with limits given in y, and rotating about the y-axis with a function that requires recognizing a perfect square trinomial to simplify the integral. By the end of this video, you will understand how to set up and evaluate surface area of revolution integrals confidently, regardless of which axis you are rotating around or which variable your function is given in. For more video lessons and additional resources, visit www.xomath.com.