the small strip width dx as indicated will have volume when revolved about x axis given by
#dV = pi( y_2^2 - pi y_1^2) dx#
where #y_2 = 20/(1+x^2)# and #y_1 = 2#
so
#dV = pi( y_2^2 - y_1^2) dx#
#= pi( (20/(1+x^2))^2 - 2^2) dx#
#V = pi int_{-3}^3 dx qquad (20/(1+x^2))^2 - 4 pi int_{-3}^3 dx qquad star#
if we do the LHS of #star# first as an indefinite integral just to keep it a bit cleaner:
#V = pi int dx qquad (20/(1+x^2))^2 #
#x = tan psi, dx = sec^2 psi d psi#
#V =400 pi int dpsi qquad sec^2 psi(1/(1+tan^2 psi^2))^2 #
# =400 pi int dpsi qquad cos^2 psi #
from the double angle formula #cos 2 psi = 2 cos^2 psi - 1#
# =200 pi int dpsi qquad cos 2 psi + 1 #
# =200 pi ( 1/2sin 2 psi + psi + C )#
#sin 2 psi = 2 cos psi sin psi#
# =200 pi ( sin psi cos psi + psi + C )#
see this for converting back
# =200 pi ( x/sqrt(1+x^2)*1/sqrt(1+x^2) + arctan x + C )#
# =200 pi ( x/(1+x^2) + arctan x + C )#
the RHS of #star# amounts to #-4pix#, so reinstating the integration interval we have
#V = pi [200 ( x/(1+x^2) + arctan x) -4x]_{-3}^3 #
# = pi(96 +400 arctan (3))#