Wednesday, March 25, 2009

Differential of Arc Length

I was trying to derive the formula for obtaining the surface area of a sphere, but I kept making mistakes whereby I chopped the surface in dx or dy. When you calculate the arc length of a curve or the surface area of a volume, you have to chop it in lengths of the slope, not the width or the height.


To compute the length of an arc of a curve, we need to define the differential of arc length. If we zoom in on a small section of a curve, we find that the length of a curve s is the length of the slope of a right triangle whose base is dx and whose height is dy.


Figure: Differential of Arc Length

Therefore,


Figure 2: Differential of Arc Length

Surfaces of Revolution


Using the differential of arc length, we can find the area of the surface generated by rotating a curve C about the x-axis or y-axis. When a curve of function y=f(x) is rotated about the x-axis, the surface area of the generated solid can be found with the following formula:


Formula for Surface Area when rotated about x-axis

When a curve of function y=f(x) is rotated about the y-axis, the surface area of the generated solid can be found with the following formula:


Formula for Surface Area when rotated about y-axis

No comments:

Post a Comment

About This Blog

KBlog logo This blog is about current events and issues concerning general population. Thanks for visiting the blog and posting your comments.

© Contents by KBlog

© Blogger template by Emporium Digital 2008

Followers

Total Pageviews

icon
Powered By Blogger