Today we will derive the surface area formula for a sphere using integral calculus. A sphere with radius r has surface area A equals 4 pi r squared. We'll use the method of revolution to derive this formula step by step.
To derive the sphere surface area, we start with a semi-circle defined by the equation y equals square root of r squared minus x squared, where x ranges from negative r to positive r. When we revolve this semi-circle around the x-axis, it creates a complete sphere.
The surface area of revolution around the x-axis is given by the integral formula: A equals the integral from a to b of 2 pi y times the square root of 1 plus dy dx squared, dx. For our semi-circle y equals square root of r squared minus x squared, we need to find the derivative dy dx. Using the chain rule, dy dx equals negative x divided by square root of r squared minus x squared.
Now we calculate the square root term. We have 1 plus dy dx squared equals 1 plus x squared over r squared minus x squared. This simplifies to the square root of r squared over r squared minus x squared, which equals r over square root of r squared minus x squared. When we substitute this into our surface area formula, the square root terms cancel out, leaving us with the integral from negative r to r of 2 pi r dx.
Now we evaluate the definite integral. The integral of 2 pi r from negative r to r equals 2 pi r times the integral of dx from negative r to r. This gives us 2 pi r times x evaluated from negative r to r, which equals 2 pi r times r minus negative r, or 2 pi r times 2r, which simplifies to 4 pi r squared. Therefore, we have successfully derived that the surface area of a sphere with radius r is A equals 4 pi r squared.