I’m doing an assignment on “applications of integration in the real world both from now, and int he beginning of the 1900’s”. I can’t seem to find much. Would someone be so kind as to point me in the right direction?
Thanks,
-Outrider
shooting off rockets, plotting orbits, lobbing missiles and other projectiles at people spring to mind
I’m actually looking for something a bit more specific. I’m interested in applications of integration, not just calculus as a whole. Also, what did people use integration for in the early 1900s?
well, this is a bit of a WAG, but I would suspect that integration has been used in navigation.
In many situations, it is (relatively) easy to determine instantaneous acceleration, but much harder to determine position or velocity. By integrating all of the instantaneous accelerations over time, and adding in the initial velocity, you can assess velocity. Then by integrating velocity, and adding initial position, you can figure out your position. And in navigation, figuring out your position is half the battle.
Yup, this is the principle by which “inertial navigation” works. This technique is heavily used in submarines, which (for obvious reasons) cannot obtain reference fixes while submerged. It’s not used as much for navigation in aircraft or surface ships because radionavigation from surface (Loran) or satellite (GPS) radio beacons is more accurate. Submarines cannot access these signals because water is opaque to radio waves (except at extremely low frequencies). Radionavigation does NOT require calculus, just straightforward geometry.
As an electrical engineer, I use integration occasionally to solve equations that describe electronic circuit behaviour. This usually involves math beyond integral calculus, such as differential equations. Here integration is just one of many tools used to figure out what you need to.
It’s a little like asking how often you use “multiplication” when working with algebra. You multiply equations by variables constantly (no pun intended :)) when solving algebraic equations, but you don’t often end up multiplying two numbers by hand on a piece of paper to solve anything. It’s just one of many tools used to find the answer your looking for.
Most engineering involves using integration as one of many math tools to solve equations. Others include more basic operations such as addition, multiplication, logarithms, exponentiation; and more advanced operations such as differential equations, Laplace and Fourier transforms, Taylor expansions, etc.
Arjuna34
Thank you for the input on some modern applications of integration. I still don’t know the answer to the other half of the question, however. I’ve spent a good deal of time pounding Google and Altavista, but there’s really no mention of what people used calculus for in the 1900s. Does anyone have an educated guess?
-Outrider
Might I just step in here a moment and remind everyone that they should be doing their own homework, and not do other folks’ work for them. I think we can give Outrider some ideas, like telling him to look up “inertial guidance” for instance, but let him fill in the details.
Oh, and by the way, Newton originally invented fluxionic calculus (what we call just plain “calculus” nowadays) to solve some problems in his gravitational theory, so that’s a start.
I know that some devices based on gyrocscopes (such as the gyrocompass) were developed in the early 1900’s. I also know that gyroscopes have something to do with inertial navigation. Can anyone flesh out the connection here? Is integration used in the principles of the gyroscope?
A semi-WAG for integration specifically, calculating the volumes of complex but well-defined shapes. For example, how much cement do you need for the Hoover dam?
I bet Nikola Tesla used integration to figure out how much power he could get out of his Death Ray.
By “Death Ray”…you mean the Tesla coil?
Outrider, I think the use of “inertial navigation” was a mistake. I think KellyM is thinking of old fashioned “Ded (for deduced) reckoning”.
Now I’m confused. Take a look at this clip from Britannica:
“There are various ways by which the position of a vessel at sea can be determined. In cases where external references such as stars or radio and satellite beacons are unavailable or undetectable, inertial navigation, which relies on a stable gyroscope for determining position, is commonly employed. It is far more accurate than the long-used technique of dead reckoning, which is dependent on a knowledge of the ship’s original position and the effects of the winds and ocean currents on the vessel.”
It actually sounds like both of these can use integration in some way. Can anyone clarify?
No, I meant what I said. Inertial nagivation is the process of navigating based on inertial data and is what is being performed by an inertial guidance unit. Inertial navigation is simply the use of double integration of the signal from precise accelerometers to determine the continuous position of the vessel. The accuracy of inertial guidance units is limited by the accuracy of the accelerometers. Small errors accumulate over time, causing loss of accuracy of fix, which is why inertial guidance units are periodically reset in use whenever the vessel gets an accurate fix from another source, such as Loran, GPS, or merely passing a known position such as a hangar or runway intersection. Since GPS is “generally available” to sea and air vessels, IG is a backup technology in such craft. Its main use as a primary navigational technology is in submarines (which do not have easy access to GPS signals) and in spacecraft.
Dead reckoning is first-order integration of velocity as opposed to second-order integration of acceleration. It is a similar concept, just less refined. The main problem with dead reckoning is obtaining accurate velocity information on a continuous basis. Inertial systems provide the accurate velocity information.