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2. In issue 3 of Derivative Girl, Darth Integrator has constructed an "infinite evil horn". The inside of
the horn can be found by rotating the region bounded by y = =, = = 2, and y = 0 around the
r-axis. Here, x and y are measured in meters. This horn will make people forget the difference
between derivation and integration.
(a) Derivative Girl plans to stuff the horn with gunk to fill it up, preventing Darth Integrator from
blowing it. How much gunk will she need? Derivative Girl can stuff the horn with 3 cubic
meters of gunk per day. How long will it take her to completely fill the horn?
(b) The surface area resulting from rotating a non-negative function, f(x), from a to b, around the
r-axis is given by / 2x(f(x))v1+ (f'(x))2 dr. We can use this formula even if the integral
is improper. Using this, give an integral that expresses the surface area of the infinite evil horn.
(c) Before using his evil horn, Darth Integrator instructs his minions, the Riemann Sum Brothers,
to paint the outside of the horn a fancy color. Darth Integrator has no shortage of buckets of
fancy color paint, but the brothers can only paint 50e square meters per day. Will Derivative
Girl finish filling the horn in time? If yes, can the brothers paint faster so that she will not? If
no, can she fill faster so that she'll be done before they are?

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