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Applications of Integration
By: Cecilia Guo and Fazena Bacchus
Introduction
1. Applications of the Indefinite Integral shows how to find displacement (from velocity) and velocity (from acceleration) using the indefinite integral. In
primary school, we learned how to find areas of shapes with straight sides (e.g. area of a triangle or rectangle). But how do you find areas when the sides
are curved? We'll find out how in:
2. Area Under a Curve and 3. Area Between 2 Curves
4. Volume of Solid of Revolution explains how to use integration to find the volume of an object with curved sides, e.g. wine barrels.
5. Centroid of an Area means the center of mass. We see how to use integration to find the centroid of an area with curved sides.
6. Work by a Variable Force shows how to find the work done on an object when the force is not constant. This section includes Hooke's Law for springs.
7. Electric Charges have a force between them that varies depending on the amount of charge and the distance between the charges. We use integration to
calculate the work done when charges are separated.
8. Average Value of a curve can be calculated using integration.
Head Injury Criterion is an application of average value and used in road safety research.
9. Force by Liquid Pressure varies depending on the shape of the object and its depth. We use integration to find the force.
In each case, we solve the problem by considering the simple case first. Usually this means the area or volume has straight sides. Then we extend the
straight-sided case to consider curved sides. We need to use integration because we have curved sides and cannot use the simple formulas any more.
Fundamental Theorem of Calculus
We are all used to evaluating definite integrals without giving the reason for the procedure much thought. The definite integral is
defined, however, not by our regular procedure but rather as a limit of Riemann sums. We often view the definite integral of a
function as the area under the graph of the function between two limits. It is not intuitively clear, then, why we proceed as we do in
computing definite integrals. The Fundamental Theorem of Calculus justifies our procedure of evaluating an antiderivative at the
upper and lower limits of integration and taking the difference.
Let f be continuous on [a,b]. If F is any antiderivative for f on [a,b], then:
Again, Let f be a continuous real-valued function defined on a closed interval [a, b].
Then, F is differentiable on [a, b], and for every x in [a, b],
The operation is a definite integral with variable upper limit, and its result F(x) is one of the infinitely many antiderivatives of f.
Let f be a real-valued function defined on a closed interval [a, b]. Let F be a function such that, for all x in [a, b],
Then, for all x in [a, b], and
History of Integration
Greek geometers are credited with a significant use of infinitesimals. Democritus
is the first person recorded to consider seriously the division of objects into an
infinite number of cross-sections, but his inability to rationalize discrete crosssections with a cone’s smooth slope prevented him
from accepting the idea. At approximately the same time, Zeno of Elea discredited infinitesimals further by his articulation of the
paradoxes which they create.
Antiphon and later Eudoxus are generally credited with implementing the method of exhaustion, which made it possible to compute
the area and volume of regions and solids by breaking them up into an infinite number of recognizable shapes. Archimedes
developed this method further, while also inventing heuristic methods which resemble modern day concepts somewhat. It was not
until the time of Newton that these methods were made obsolete. It should not be thought that infinitesimals were put on rigorous
footing during this time, however. Only when it was supplemented by a proper geometric proof would Greek mathematicians accept
a proposition as true.
History of Integration
In the third century Liu Hui wrote his Nine Chapters and also Haidao suanjing, which dealt with using the Pythagorean theorem
(already stated in the
Nine Chapters), known in China as the Gougu theorem, to measure the size of things. He discovered the usage of Cavalieri’s
principle to find an accurate formula for the volume of a cylinder, showing a grasp of elementary concepts associated with the
differential and integral calculus.
Indian mathematicians produced a number of works with some ideas of calculus. The formula for the sum of the cubes was first
written by Aryabhata circa 500 AD, which was an important step in the development of integral calculus.
Around 1000 AD, Ibn al-Haytham (known as Alhazen in the West), an Iraqi mathematician working in Egypt, was the first
mathematician to derive the formula for the sum of the fourth powers. In turn, he developed a method for determining the general
formula for the sum of any integral powers, which was fundamental to the development of integral calculus.
In the 17th century, Pierre de Fermat, among other things, is credited with an ingenious trick for evaluating the integral of any
power function directly, thus providing a valuable clue to Newton and Leibniz in their development of the fundamental theorems of
calculus.
At around the same time, there was also a great deal of work being done by Japanese mathematicians, particularly Kowa Seki. He
made a number of contributions, namely in methods of determining areas of figures using integrals, extending the method of
exhaustion. While these methods of finding areas were made largely obsolete by the development of the fundamental theorems by
Newton and Leibniz, they still show that a sophisticated knowledge of mathematics existed in 17th century Japan
Here are the “real world
problems” and “methods”:
To Apply or how to Apply that is the
question!
-Mathspearean
Question 1
Work by a Variable Force
• The work (W) done by a constant force (F) acting on
a body by moving it through a distance (d) is given
by:
• W=F×d
• If the force varies (e.g. compressing a spring) we
need to use calculus to find the work done.
• If the force is given by F(x) (a function of x) then the
work done by the force along the x-axis
from a to b is:
• The force (F) that it takes to stretch (or compress) a
spring x units from its normal length is proportional
to x.
• F = kx
• We can find the spring constant k from observing
what force gives what stretch for each spring.
Find the work done on a spring when you
compress it from its natural length of 1 m to a
length of 0.75 m if the spring constant is k = 16
N/m.
• Given:
F = 16x
Apply integration…
Question 2
Many solid objects, especially those made on a lathe,
have a circular cross-section and curved sides. In this
section, we see how to find the volume of such objects
using integration.
• A watermelon has an ellipsoidal shape with major axis 28 cm
and minor axis 25 cm. Find its volume.
• Historical Approach: Before calculus, one way of
approximating the volume would be to slice the watermelon
(say in 2 cm thick slices) and add up the volumes of each
slice using V = πr2h.
• We are told the melon is an ellipsoid.
We need to find the equation of the
cross-sectional ellipse with major axis
28 cm and minor axis 25 cm.
• We use the formula
(from the section on ellipses):
• where a is half the length of the major axis and b is half the
length of the minor axis.
• For the volume formula, we will need the expression
for y2 and it is easier to solve for this now (before substituting
our a and b).
Since a = 14 and
b = 12.5, we have:
ANSWER:The watermelon's total volume is 2 × 4580.65 = 9161 cm3 or 9.161L
Question #3:
The average value of the function y = f(x) from x =
a to x = bis given by…
-The temperature T (in °C) recorded
during a day followed the curve
T = 0.001t4 − 0.280t2 + 25
-where t is the number of hours from
noon (-12 ≤ t ≤ 12)
What was the average temperature during the day? First,
we consider the graph of the situation and estimate that
the average should be around 14 to 16 degrees.
Question #4:
The force between charges is proportional to the product
of their charges and inversely proportional to the square
of the distance between them.
So we can write:
~Where q1 and q2 are in coulombs (C), x is in meters,
the force is in Newtons and k is a constant, k = 9 × 109.
~It follows that the work done when electric charges
move toward each other(or when they are separated)
is given by:
An electron has a 1.6 × 10-19 C negative charge.
How much work is done in separating two electrons
from 1.0 pm to 4.0 pm?
In this example,
• a = 1 × 10-12 m
• b = 4 × 10-12 m
• k = 9 × 109
• q1 = q2 = 1.6 × 10-19 C
• So we have:
References
Math Type
http://www.intmath.com/applicationsintegration/applications-integrals-intro.php