How the Area of a Disk Grows

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An increase in the radius of a circular disk leads to a corresponding increase
in the area of the disk. Geometrically,
is the red annulus. Slicing and then unrolling this annulus would yield a shape close to a rectangle with base
and height
. This gives the approximation
. Using the slider, notice this approximation is quite accurate for
close to zero. Graphing the area
as a function of the radius
, the ratio
is the slope of a secant line to the curve. Dividing both sides of
by
implies the slope
may be approximated by the circumference
.
Contributed by: Jim Brandt (September 2014)
Open content licensed under CC BY-NC-SA
Snapshots
Details
Taking the limit as approaches zero, the approximation
becomes
, which expresses the derivative of
with respect to
.
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