# (a) Find the volume V of the solid generated when the region bounded by is revolved about the y-axis. (b) Find `lim_(b->+oo)V` `` I found part (a) to be `pi(arctan(b^2)-pi/4)` by using...

(a) Find the volume V of the solid generated when the region bounded by

is revolved about the y-axis.

(b) Find `lim_(b->+oo)V` ``

I found part (a) to be `pi(arctan(b^2)-pi/4)` by using the shell method.

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Expert Answers

mathsworkmusic | Certified Educator

Taking principal value solutions of arctan, or the inverse tangent function, that is, solutions in the range `(-pi/2,pi/2)` , arctan `b^2` tends to `pi/2` as `b` tends to infinity.

Formally,

`lim_(b->oo) arctan(b^2) = pi/2`

Therefore, if from part a)

`V = pi(arctan(b^2)-pi/4)`

then

b) `lim_(b->oo) V = lim_(b->oo) pi(arctan(b^2) - pi/4) = pi(pi/2 - pi/4) = pi(pi/4) = pi^2/4`

**answer**