Problem 45 - Essential Calculus

Question

What precalculus formula and representative element are used to develop the integration formula for the area of a surface of revolution?

Step-by-Step Solution

Answer

The precalculus formula used to develop the integral formula for the area of a surface of rotation is the formula for the circumference of a circle, \(2πr\), and the representative element is an infinitesimally thin slice of the function with a width \(dx\), treated as a cylindrical element.

Step 1: Identify Precalculus Formula

The precalculus formula used is the formula for the circumference of a circle, which is \(2πr\), where \(r\) is the distance from the axis of rotation to the graph.

Step 2: Identify Representative Element

The representative element in this case is an infinitesimally thin slice of the function width \(dx\) that can be treated as a small cylinder. The surface area of this slice is \(2πrh\), where \(r\) is the function \(f(x)\) at a given point and \(h\) is the slice's width, or \(dx\).

Step 3: Combine to Form Integral

To find the total area of the surface of revolution, the surface areas of all the small slices must be added up. This is done by integrating the surface area formula \(\int 2πf(x)dx\)