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"The region enclosed by the curvesand
is roated about the line
. Fint the volume of the resulting solid" -James Stewart
answer:
what I did:
difference of shells formula is
so in this problemand
. The intersections are
. I get confused because it's not being rotated around the y-axis but
I triedbut I end up with a negative number.
The x term represents the distance between an arbitrary x and the axis of rotation.
If the solid is rotated about the yaxis the x term is just x.
In your example the distance between x and the axis of rotation x = 1 is
1-x. so replace x with 1-x--- you obtain (1-x)*(sqrt(x)-x^3) for the integrand.
A more general form of the shell method is 2pint{D(x)*(f-g)dx}
Where D(x) is as mentioned the distance between an arbitrary x and the axis of rotation
