Applications of the Integral
Flow rates and volumes
One of the most commonly known applications of the integral is calculation of volume and area of irregular
shapes and surfaces. Most intermediate calculus texts incorporate these applications in the commonly used
filling or draining a tank with an unconventional shape. However, this isn't very often used in engineering
applications, as tanks are usually shaped fairly normally. Finding volumes of small unconventionally shaped
objects is a simple matter of water displacement, but scientists and engineers sometimes need to find the
volume of a larger shape that cannot be easily submerged and displacement measured. In these cases, integral
calculus comes to the rescue! By either manually or digitally recreating the object (to scale) in a graphical
interface, then using this representation to make a linear and/or planar regression of the object, these
regressions can be integrated and the volume found.
Along with finding volumes and areas of strangely shaped (as far as geometry goes) objects, integrals can
also be used to calculate flow rates of fluids in a variety of places. Using field vectors to represent the
flow at any given point in a river, for instance, an integral over an area can tell how much water goes through
that area over a time interval. In air flow, these vector fields and integrals can be used to model the speed and
and pressure of the air around an airplane wing or through a respirator.
For more in-depth information on the application of integrals to model
flow rates, visit this page.
Electromagnetic Fields and Work
As mentioned above, vector fields are a branch of mathematics wherein physicists can use integrals to solve
meaningful problems. These vector fields can represent gravitational fields and electromagnetic fields, and
with them, physicists can model the effect of these fields on particles with mass or charge respectively.
With integration, work done on a particle or particles can be calculated by doing line integrals. These line
integrals take the distance traveled and the magnitude of the field and particle to calculate work (W=Fd) and are used especially
for electromagnetic problems. For more in-depth information on field vectors and line integrals
visit this page.
Optimizations
Optimization problems are common in mathematical settings, as early as algebra classes. Even when the word
optimization is not used, many math problems are focused on finding the most efficient or productive way to do
a given thing. With integration, more complex problems can be optimized. For example, using integration, a
farmer may want to maximize flow rate in his canal, but wants to minimize the amount of concrete needed to pour
the base of it. Using integration techniques on vector fields, he is able to maximize flow with the
geometric shape used as the cross-sectional area and use differentiation to minimize the perimeter.
Another example of where integrals are applied to optimization problems is in a free-fall type setting in
the earth's atmosphere. Looking to minimize terminal velocity with a three dimensional shape, taking a
surface integral can show how much surface area comes in contact with air, and how the vector flow of air
would shape itself around such a surface. This integration would help optimize the shape of a parachute or
other flying device.
For more information on engineering applications and optimizations using integrals, visit this page.
Onward! To your Applets!