A right circular cylinder has a radius of 40 cm and a height of 100 cm. Use differentials to estimate the change in volume of the cylinder if its height and radius are both increased by 1 cm. Give an exact answer.

The change in volume is equal to pi*81cm^3

Step-by-step explanation:

The volume of a cilinder can be obtained by the integral:

where i stands for initial values, and f for final values, a is the angle, that goes from 0  to 2*pi.,

the initial values are:

R = 40cm, Z = 100cm

the final ones are:

R = 41cm, Z = 101 cm

that integral is equal to:

((r^2)/2)*z*a) valuated between the initial and final values, exept for the angle a.

Change in volume  = (((41cm^2) - (40cm)^2)/2)*(101cm - 100cm)*(2*pi - 0)

=2*pi*1cm*40.5cm^2 = pi*81cm^3

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