## You are here

Homeintegration with respect to surface area on a helicoid

## Primary tabs

# integration with respect to surface area on a helicoid

To illustrate the result derived in example 3, let us compute the area of a portion of helicoid of height $h$ and radius $r$. (This calculation will tell us how much material is needed to make an Archimedean screw.) The integral we need to compute in this case is

$A=\int d^{2}A=\int_{0}^{{h/c}}\int_{0}^{r}\sqrt{c^{2}+u^{2}}\>du\,dv=$ |

$\int_{0}^{{h/c}}\left({1\over 2}r\sqrt{c^{2}+r^{2}}+{c^{2}\over 2}\log\left\{% \frac{r}{c}+\sqrt{1+\left(\frac{r}{c}\right)^{2}}\right\}\right)\>dv=$ |

$\frac{rh}{2}\sqrt{1+\left(\frac{r}{c}\right)^{2}}+{ch\over 2}\log\left\{\frac{% r}{c}+\sqrt{1+\left(\frac{r}{c}\right)^{2}}\right\}$ |

As a second illustration, let us compute the second moment of a helicoid about the axis of rotation. In mechanics, this would be called the moment of inertia of the helicoid and determines how much energy is needed to make the screw rotate. This is determined as follows:

$\displaystyle\int(x^{2}+y^{2})d^{2}A=$ | $\displaystyle\int_{0}^{{h/c}}\int_{0}^{r}u^{2}\sqrt{c^{2}+u^{2}}\>du\,dv=$ | |||

$\displaystyle\int_{0}^{{h/c}}\left(\frac{r(2r^{2}+c^{2})}{8}\sqrt{c^{2}+r^{2}}% -\frac{c^{4}}{8}\log\left\{\frac{r}{c}+\sqrt{1+\left(\frac{r}{c}\right)^{2}}% \right\}\right)\,dv=$ | ||||

$\displaystyle\frac{rh(2r^{2}+c^{2})}{8}\sqrt{1+\left(\frac{r}{c}\right)^{2}}-% \frac{hc^{3}}{8}\log\left\{\frac{r}{c}+\sqrt{1+\left(\frac{r}{c}\right)^{2}}\right\}$ |

Quick links:

Type of Math Object:

Example

Major Section:

Reference

Groups audience:

## Mathematics Subject Classification

28A75*no label found*

- Forums
- Planetary Bugs
- HS/Secondary
- University/Tertiary
- Graduate/Advanced
- Industry/Practice
- Research Topics
- LaTeX help
- Math Comptetitions
- Math History
- Math Humor
- PlanetMath Comments
- PlanetMath System Updates and News
- PlanetMath help
- PlanetMath.ORG
- Strategic Communications Development
- The Math Pub
- Testing messages (ignore)

- Other useful stuff
- Corrections