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Surface of revolution


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Surface of Revolution


Equations are omitted for technical reasons - download the original pdf

Let us suppose we have a curve given by y = f(x). [Diagram goes here - download the original to see it.] When we rotate this curve around the x-axis, between specific points a and b, we obtain a surface of revolution: [Diagram goes here - download the original to see it.] It is a mathematical problem to discover a formula for the area of this surface of revolution. Our approach is the usual one of dividing the surface up into segments and approximating each segment. Firstly, let us divide the surface into cylindrical like portions of area [Diagram goes here - download the original to see it.]. [Diagram goes here - download the original to see it.] We now observe that each segment is approximately a segment of a cone. [Diagram goes here - download the original to see it.] If we now imagine taking this segment of the cone, making a vertical cut in one side of it and opening it out, we would obtain approximately a trapezium. [Diagram goes here - download the original to see it.] So the area of the curved surface segment on our original surface of revolution is approximately equal to the area of a curved surface of a right circular cone. The area of a curved surface of a right circular cone is given by: [Equation goes here - download the original to see it.] Here r is the radius of the smaller circle and R is the radius of the larger circle. [Equation goes here - download the original to see it.] Now the radius of the curve segment is given by [Diagram goes here - download the original to see it.] and the height of the trapezium is [Diagram goes here - download the original to see it.] length of line segment. Hence each area segment of our surface of revolution is [Equation goes here - download the original to see it.] In the limit, as [Diagram goes here - download the original to see it.] [Diagram goes here - download the original to see it.] The whole surface of revolution is given by the integral of the surface elements. [Diagram goes here - download the original to see it.] But ds, the infinitesimal arc length, is given by [Equation goes here - download the original to see it.] Hence, the surface of revolution is: [Diagram goes here - download the original to see it.]
Contents of
Surface of revolution

1 Surface of Revolution
2 Example Surface of Revolution

Related articles: (1) Arc length of a curve in parametric form, (2)