Changes

== Example ==

== Example ==

Consider the region in the first quadrant that has an upper bound of <math>y = \sqrt 2</math> and a lower bound of <big><math>y = (\sec{x})(\tan{x})</math></big>, and bounded on the left side by the ''y-axis''.  Find the volume of the solid formed by rotating the region about the line <math>y = \sqrt 2</math>.

Consider the region in the first quadrant that has an upper bound of <math>y = \sqrt 2</math> and a lower bound of <big><math>y = (\sec{x})\,(\tan{x})</math></big>, and bounded on the left side by the ''y-axis''.  Find the volume of the solid formed by rotating the region about the line <math>y = \sqrt 2</math>.

[[Calculus]] provides an elegant way to determine the volume of this solid.  First, find where the curves intersect in order to ascertain the end-point of the integration.  The boundaries intersect where

[[Calculus]] provides an elegant way to determine the volume of this solid.  First, find where the curves intersect in order to ascertain the end-point of the integration.  The boundaries intersect where

The next insight is to express ''r'' in terms of ''x''.  The variable ''r'' is the distance of the boundary from the axis about which it is rotated:

The next insight is to express ''r'' in terms of ''x''.  The variable ''r'' is the distance of the boundary from the axis about which it is rotated:

:<math>r = \sqrt 2 - \sec{x}\,\cos{x}</math>

:<math>r = \sqrt 2 - \sec{x}\,\tan{x}</math>

[[category:mathematics]]

[[category:mathematics]]

[[category:calculus]]

[[category:calculus]]