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== 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 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
<math>y = \sqrt 2 = (\sec{x})(\tan{x})</math>
or
<math>\sqrt 2 = \frac{\sin{x}}{\cos^2{x}}</math>
This is best solved by trial-and-error. Since <math>\sin{\frac{\pi}{4}} = \cos{\frac{\pi}{4}}=\frac{\sqrt 2}{2}</math>, that is the solution.
[[category:mathematics]]
[[category:mathematics]]
[[category:calculus]]
[[category:calculus]]