Changes

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 ==

:<math>\int_0^\frac{\pi}{4}(\sec{x}\,\tan{x})^2\,dx</math>

:<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^3{x}})\,dx</math>

:<math> = \int_0^\frac{\pi}{4}\sin{x}\,(\frac{\sin{x}}{\cos^4{x}})\,dx</math>

:<math> = \frac{\sin{x}}{2\cos^2{x}} - \int_0^\frac{\pi}{4}\frac{sec{x}}{2},\,dx</math>

:<math> = \frac{\sin{x}}{3\cos^3{x}} - \int_0^\frac{\pi}{4}\frac{\sec^2{x}}{3},\,dx</math>

Recall that:

Recall that:

:\int{sec{x},\,dx = tan{x}</math>

:<big><math>\int\sec^2{x}\,dx = \tan{x}</math></big>

and the solution becomes obvious.

and the solution to the overall integral is easy to obtain.

[[category:mathematics]]

[[Category:Mathematics]]

[[category:calculus]]

[[Category:Calculus]]