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Calculus II (Notes) / Parametric Equations and Polar Coordinates / Surface Area with Polar Coordinates   [Notes] [Practice Problems] [Assignment Problems]

Calculus II - Notes

Internet Explorer 10 & 11 Users : If you have been using Internet Explorer 10 or 11 to view the site (or did at one point anyway) then you know that the equations were not properly placed on the pages unless you put IE into "Compatibility Mode". I beleive that I have partially figured out a way around that and have implimented the "fix" in the Algebra notes (not the practice/assignment problems yet). It's not perfect as some equations that are "inline" (i.e. equations that are in sentences as opposed to those on lines by themselves) are now shifted upwards or downwards slightly but it is better than it was.

If you wish to test this out please make sure the IE is not in Compatibility Mode and give it a test run in the Algebra notes. If you run into any problems please let me know. If things go well over the next week or two then I'll push the fix the full site. I'll also continue to see if I can get the inline equations to display properly.

 Surface Area with Polar Coordinates

We will be looking at surface area in polar coordinates in this section.  Note however that all we’re going to do is give the formulas for the surface area since most of these integrals tend to be fairly difficult.

 

We want to find the surface area of the region found by rotating,

 

 

about the x or y-axis.

 

As we did in the tangent and arc length sections we’ll write the curve in terms of a set of parametric equations.

 

 

 

 

If we now use the parametric formula for finding the surface area we’ll get,

 

 

where,

                                

 

Note that because we will pick up a  from the ds we’ll need to substitute one of the parametric equations in for x or y depending on the axis of rotation.  This will often mean that the integrals will be somewhat unpleasant.


Calculus II (Notes) / Parametric Equations and Polar Coordinates / Surface Area with Polar Coordinates    [Notes] [Practice Problems] [Assignment Problems]

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