Volumes of Revolution

**jhomie** · November 14th 2008, 12:28 AM

I have this question which is rather interesting and quite hard IMO.

I have to find the volumes of revolution generated by the regions A and B when they are each rotated about: the x axis and the y axis

If the equation was

from a to b,then it means I need to find the volume of revolution for:

x = a to b, rotated around the x axis, which would be:
y = a to b, roated around the y axis, which would be:

are those right?

I also have to find this though and this is what I need help with:

x = a to b, rotated around the y axis
y = a to b, roated around the x axis

I dont know what to do for this though

**shawsend** · November 14th 2008, 02:27 PM

Originally Posted by jhomie

I have this question which is rather interesting and quite hard IMO.

I have to find the volumes of revolution generated by the regions A and B when they are each rotated about: the x axis and the y axis

If the equation was

from a to b,then it means I need to find the volume of revolution for:

x = a to b, rotated around the x axis, which would be:
y = a to b, roated around the y axis, which would be:

Those are correct. The other two involve volumes using cylindrical shells which have the formula $V=2\pi\int_a^b xf(x)dx$ or $2\pi\int_a^b yg(y)dy$ so:

$V_y=2\pi\int_a^b x^{n+1}dx$

$V_x=2\pi\int_a^b y^{1+1/n}dy$

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