I need help on 2 calculus questions, Volume of A Solid of Revolution?

For #2 :

One can do shells or washers about both x and y rotations

see here for some links :

http://www.khanacademy.org/math/calculus/solid_rev...

and here :

http://www.khanacademy.org/math/calculus/solid_rev...

One normally picks the method which uses the simplest form of y = f(x) or x = f(y )

Practise with both will tell you which gets to an answer, faster...

However.. Note this carefully . Both give EXACTLY the same answer, if you use the correct x and y limits... so except for the difficulty of getting f(x) = y or x = f(y) , both are good.

( Common beginner error : Using x limits for a dy integration )

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For #1

When we integrate with disks ... You are correct ... we add up a whole bunch of tiny disks to get the final volume ..

With the shell method, we are adding up a whole bunch of thin cylindrical shells ..

A possible model is to take several empty toilet paper cardboard .centers , cut them lengthwise, and stack them around each other, to create the shells

My teaching method, was to carefully draw one rotating rectangle, then sketch just one shell.. Its a hollow cylindrical shell.. similar to the cardboard from a roll of toilet paper ,

usually with height equal to the difference of the top and bottom functions, and radius of x

NOW, cut it open ( in your mind, of course )

and flatten it out , as best you can( again in your mind )

Its now a rectangular slab :

its thickness is the small dx

its height is once again (top f(x) - bottom f(x) )

and its width is the circumference of the original shell :

width = C = 2πr = 2πx

and we integrate ∫2πx(top - bottom )dx from the smallest x to the largest x

Good Luck !