Microworld: Applications
of Integration
Click the Hyperlink
above to visit the Microworld.
Author:
Ravinder
Kumar
This
9-page microworld explores arc length of a curve, area under a curve, and
surface area and volume of revolution. For simplicity we explore only those
surfaces of revolution that can be obtained by revolving a curve about x-axis.
Arc
length, area, surface area, and volume can be found by dividing the arc, region,
or solid into tiny portions in Riemannian spirit. You will be living in Riemannian
spirit as you conduct explorations on the following interactive pages.
The
theory will be briefly explained on the help pages that can be viewed by pressing
the button “math for this page”. Often an example or two may be used to explain
the theory. When a page of the microworld contains a button named “instructions”,
you can press it to view instructions for using the interactivity of the page
in order to make explorations.
Seeds
for the ideas of integration that lead up to finding area and volume were
sown much earlier than the advent of calculus. Historically,
the concept of infinitesimal sums, in the context of determining area and
volume, as well as expansion of series occurred much earlier. Based on the
Greek method of “exhaustion” resulting from the contributions of Leucippus,
Democritus and Antiphon between 450 BC and 370 BC and put on a scientific
basis by Eudoxus about 370 BC, Archimedes (around 225 BC) determined the area
of a segment of a parabola, circle, surface area and volume of a cone and
sphere etc.
According
to the exhaustion method, a given area, for example, was thought of a sequence
of expanding areas leading (approaching) up to the given area. More explicitly,
consider regular polygons of sides 4, 8, 16, 32, 64, 128, … inscribed in a
circle. Area of each regular polygon expands the area of the preceding polygon,
and is closer to the area of the circle than before.
Traces
and results of differential and integral calculus are also found in other
cultures. For example, Bhaskara II, a well known Indian mathematician of middle
ages wrote a math book Lilavati in 1150 A.D. in the memory of his daughter.
In this book and his other works one can find important concepts and results
from differential and integral calculus. Work of many Indian mathematicians
of middle ages was translated in the Arab world and China, mainly because
of the efforts of excellent and learned travelers from these countries to
India.
It
was much later in the seventeenth century that Newton and Leibnitz actually
started relating integral and differential calculus as we know today - through
the concept of an antiderivative. Nineteenth century outstanding German mathematician
Georg Friedrich Bernhard Riemann (1826-1866) seems to have put this relationship
on a rigorous footing by defining a definite integral in terms of area. His
approach certainly relies on the method of exhaustion.
This
book may be used for
- Calculus I
- Calculus II
Number
of Pages: 9
Animation: Yes
Grade Level: 11-15
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