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clean up & uniformity
In [[calculus]], solids are formed by rotating a curve around an axis and integrating to find the volume. Typically the [[integration]] is of slices cut vertically to the axis of the rotation that formed the solid. Those slices are then integrated from one end of the solid to the other.
In [[calculus]], '''solids''' are formed by rotating a curve around an axis and integrating to find the volume. Typically the [[integration]] is of slices cut vertically to the axis of the rotation that formed the solid. Those slices are then integrated from one end of the solid to the other.
== 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>.
[[category:mathematics]]
[[category:calculus]]
[[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. 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 now 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>
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}\,\tan{x}</math>
The volume then becomes:
:<math>V = \int_0^\frac{\pi}{4} {\pi}(\sqrt 2 - \sec{x}\,\tan{x})^2\,dx</math>
:<math>V = \int_0^\frac{\pi}{4} {\pi}(2 - 2\sqrt2\sec{x}\,\tan{x} + (\sec{x}\,\tan{x})^2\,dx</math>
The only challenging part of this [[integral]] is the last term, which must be [[integrated by parts]]:
:<math>\int_0^\frac{\pi}{4}(\sec{x}\,\tan{x})^2\,dx</math>
:<math> = \int_0^\frac{\pi}{4}\sin{x}\,(\frac{\sin{x}}{\cos^4{x}})\,dx</math>
:<math> = \frac{\sin{x}}{3\cos^3{x}} - \int_0^\frac{\pi}{4}\frac{\sec^2{x}}{3},\,dx</math>
Recall that:
:<big><math>\int\sec^2{x}\,dx = \tan{x}</math></big>
and the solution to the overall integral is easy to obtain.
[[Category:Mathematics]]
[[Category:Calculus]]