Thursday, February 08, 2007

6.3 Volumes by Cylindrical Shells

Ok, so in the last lesson, which Kate has failed to post (jk kate... its funny bc now I have back to back posts! lucky me.... well let's just say I'm done for the year!) we learned how to find volumes by using DISKS and WASHERS. WEll, Luckily enough for us, we now get to learn how to find Volumes by using CYLINDRICAL SHELLS!!!! I know I just made all of your lives complete by informing you of this good news.



The reason for using cylindrical shells is if we are faced with a difficult problem like finding the volume of this shape around the y-axis.













If we had to find π ∫ R^2 - r^2 dy, (using the dashed line)





It would br rather difficult to solve for x in terms of y.









So, instead we can use the bolded line, and instead find the volume of the cylindrical shell made by rotating around the y-axis.





Volume of a cylinder = 2πrh(dx)





Where......





r = x and h = top - bottom = y-values = 0 - ((x^3)-x)



Another example:

Using cylindical shells (sections parallel to the axis of rotation), find the volume of the area enclosed by y = x, y = x^2, about the y-axis

V = 2π ∫ r h (dx) on the interval a to b.


Then, r = x
h = top - bottom = x- x^2

V = 2π ∫ x(x - x^2) dx on the interval 0 to 1.


That's it. Wahooo for Calc blog.

Here is a link......

http://www.geocities.com/pkving4math2tor7/7_app_of_the_intgrl/7_03_02_finding_vol_by_using_cylind_shells.htm


I don't know WHO the Lucky person is that's Up Next....


But, It's GREY's NIGHT!!!!!!

EVERYONE, Turn ON ABC At 9 for the TIME OF YOUR LIFE. TRUST ME ON THIS ONE.





-Lauren




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