AIM OF THE EXPLORATION
ÿ Explore Integral Calculus usage to find the Volume of the solids
ÿ Identify the cross-section of the solid
ÿ Suggest an Algorithm for finding an expression for the Volume
INTRODUCTION
We learnt in the Applications of the Integral calculus to find the area under the curve.
This can be divided in following three cases:
ÿ Area below any given curve and above the X-axis
ÿ Area between the two given curves
If definite integration can be used to calculate the area of any figure in XY plane, then there must be
some way to calculate the volume of Figures in 3 Dimensional Geometry. can calculus be used for this
purpose.
Yes definitely, and that is the topic of our exploration. We will try to demonstrate the use of calculus
or Definite Integration to find the volume of certain figures having the cross-sections whose area we
already know.
We will take 4 cases:
ÿ Solid with cross-section of a square
ÿ Solid with cross-section of a equilateral triangle
ÿ Solid with cross-section of a circle
ÿ Solid with cross-section of a rectangle
To demonstrate the use of calculus, we will be taking certain examples and solving them
MATHEMATICAL DEMONSTRATION
The Simplest Case—One curve with the area entirely above the x-axis
The example of this type is the region in the first quadrant bounded by , x-axis
and .
Begin with a sketch of the problem. The dark line represents the ith rectangle.
The curve clearly passes through x=0, so x=0 is the lower bound of the definite integral.
.
The top curve is and the bottom curve is y=0 so the integral we need is
Area between two curves
the area between two curves f and g from a to b is given by
f (x) ≥ g(x) ∀ x ∈ (a,b)
This integral will always yield the ...
... middle of paper ...
...ce as seen from "above".
The area of a rectangle is base times height.
The base of our rectangle = the distance from f to g which is 2 .
The area of the cross section = 4 . The thickness is dx
The volume of our slice is 4 dx .
The required volume will be given by,
INFERENCE
Algorithm to derive an expression for the volume of the solids with known cross-sections is as follows.
ÿ The region of the base of the solid is drawn
ÿ An arbitrarily chosen slice of the solid with the desired orientation is drawn and the
thickness is denoted as dx or dy
ÿ The shape of the cross section is drawn
ÿ Measure the parameters of the solid in terms of x or y looking at the base.
ÿ Write dV , the volume of one arbitrary slice using geometry formulas.
ÿ Use the integral for the volume V, by seeing at the base to determine where the slices
start and stop.
ÿ Evaluate.
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Calculus, the mathematical study of change, can be separated into two departments: differential calculus, and integral calculus. Both are concerned with infinite sequences and series to define a limit. In order to produce this study, inventors and innovators throughout history have been present and necessary. The ancient Greeks, Indians, and Enlightenment thinkers developed the basic elements of calculus by forming ideas and theories, but it was not until the late 17th century that the theories and concepts were being specified. Originally called infinitesimal calculus, meaning to create a solution for calculating objects smaller than any feasible measurement previously known through the use of symbolic manipulation of expressions. Generally accepted, Isaac Newton and Gottfried Leibniz were recognized as the two major inventors and innovators of calculus, but the controversy appeared when both wanted sole credit of the invention of calculus. This paper will display the typical reason of why Newton was the inventor of calculus and Leibniz was the innovator, while both contributed an immense amount of knowledge to the system.
Ever wonder how scientists figure out how long it takes for the radiation from a nuclear weapon to decay? This dilemma can be solved by calculus, which helps determine the rate of decay of the radioactive material. Calculus can aid people in many everyday situations, such as deciding how much fencing is needed to encompass a designated area. Finding how gravity affects certain objects is how calculus aids people who study Physics. Mechanics find calculus useful to determine rates of flow of fluids in a car. Numerous developments in mathematics by Ancient Greeks to Europeans led to the discovery of integral calculus, which is still expanding. The first mathematicians came from Egypt, where they discovered the rule for the volume of a pyramid and approximation of the area of a circle. Later, Greeks made tremendous discoveries. Archimedes extended the method of inscribed and circumscribed figures by means of heuristic, which are rules that are specific to a given problem and can therefore help guide the search. These arguments involved parallel slices of figures and the laws of the lever, the idea of a surface as made up of lines. Finding areas and volumes of figures by using conic section (a circle, point, hyperbola, etc.) and weighing infinitely thin slices of figures, an idea used in integral calculus today was also a discovery of Archimedes. One of Archimedes's major crucial discoveries for integral calculus was a limit that allows the "slices" of a figure to be infinitely thin. Another Greek, Euclid, developed ideas supporting the theory of calculus, but the logic basis was not sustained since infinity and continuity weren't established yet (Boyer 47). His one mistake in finding a definite integral was that it is not found by the sums of an infinite number of points, lines, or surfaces but by the limit of an infinite sequence (Boyer 47). These early discoveries aided Newton and Leibniz in the development of calculus. In the 17th century, people from all over Europe made numerous mathematics discoveries in the integral calculus field. Johannes Kepler "anticipat(ed) results found… in the integral calculus" (Boyer 109) with his summations. For instance, in his Astronomia nova, he formed a summation similar to integral calculus dealing with sine and cosine. F. B. Cavalieri expanded on Johannes Kepler's work on measuring volumes. Also, he "investigate[d] areas under the curve" ("Calculus (mathematics)") with what he called "indivisible magnitudes.