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[[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
<math>y = \sqrt 2 = (\sec{x})(\tan{x})</math>
:<math>y = \sqrt 2 = (\sec{x})(\tan{x})</math>
or
or
<math>\sqrt 2 = \frac{\sin{x}}{\cos^2{x}}</math>
:<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.
This is best solved by trial-and-error. Since
The [[integral]] to find the volume of the solid must therefore be taken from ''x=0'' on one side to <math>x=\frac{\pi}{4}</math> on the other side.
:<math>\sin{\frac{\pi}{4}} = \cos{\frac{\pi}{4}}=\frac{\sqrt 2}{2}</math>,
that is the solution. The [[integral]] to find the volume of the solid must therefore be taken from ''x=0'' on one side to:
:<math>x=\frac{\pi}{4}</math>
on the other side.
We are not ready to find the volume. Note first that:
:<math> dV = {\pi}r^2dx</math>
and thus
:<math>V = \int_0^\frac{\pi}{4} {\pi}r^2\,dx</math>
[[category:mathematics]]
[[category:mathematics]]
[[category:calculus]]
[[category:calculus]]