I’m in CalcII now and for the life of me cannot wrap my head around integrals. Now we are using things like u-Substitution methods and solving the areas between 2 curves. I can understand how the equations work, but not why because I still cannot picture what an integral is or why it’s important.
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If I give you a list of numbers and ask you to add them up that’s called a sum. Now imagine I gave you a list of numbers that was infinitely long and asked you to add them up. That’s an integral.
Integrals are a supercharged multiplication technique that allows you to multiply numbers that change. Doing this manually is very difficult and requires a lot of practice.
*e.g. the area under the curve of 2×2 is a box with area 4, with integrals it’s a much fancier shape
Using your background from Calc 1 let’s first look at how a derivative works.
Imagine a car moving along a highway at a set speed. That is known as the velocity. If a red traffic light is up ahead the car will start to slow down, so there will be a change in this speed which you can feel.
We can measure this change by taking the derivative of the speed, we know this result as the acceleration of the car.
Once the traffic light is green again there is a change in the acceleration, now speeding up instead of slowing down.
We can measure this change in the acceleration by also taking its derivative, we know this result as the jerk. Which you may also feel when there are changes in acceleration.
Knowing this you can continue to take derivatives into further levels of change.
I like to imagine this as an elevator in a hotel, you’re descending lower and lower to see a specific view.
velocity -> acceleration -> jerk
The integral is the opposite of the derivative, so instead of descending these hotel floors you’re ascending them to see a more broad view instead of a specific part.
jerk -> acceleration -> velocity
You’re driving your car at 50km/h. You drive for one hour. What distance did you drive?
You’re driving your car at 40km/h. You drive for thirty minutes. Then you speed up to 50km/h, and drive for another thirty minutes. What distance did you drive this time?
(many steps later)
You’re driving your car at 40km/h. You drive for one milisecond. Then you speed up to 40.0001km/h, and drive for another milisecond. Then you speed up to… etc etc etc etc. What about _this_ time, what distance did you drive?
In the limit, as you approach an infinite number of infinitesimally small time slices, that’s the integral of speed over time.
The premise of an integral is to get information using what you know, generally the rate of change. If we know a car is accelerating 10 miles per hour, we don’t know how fast it’s going but we can figure out how much faster it has gone, and in addition, how much more MORE distance it’s covered as a result of that acceleration. If we gain the knowledge of how fast that car WAS going, we now know not only how fast it is now but can also determine the distance it traveled.
I might suck at where the words come from here, but when you take the derivative, you’re finding what measurement caused the other measurement, or what derived it, like speed causing a change in position or speed being derived from acceleration. When you integrate, you’re integrating (bringing all of these things together) to figure out what the speed or position was.
I have a hunch I’m WAAAY above the ELI5 here, but outside of word problems most of the math in this area is going to be abstract and not very intuitive.
Lots of great answers here. I’ll give you some resources instead.
Khan Academy is awesome it got me thru calc 1-4 (calc 2 and 4 two times… lol)
3blue1brown provides some great visualization of calculus concepts you may find useful.
You got this. If you fail, try again. I failed calc 2 and 4 my first time thru, I’m now a successful engineer. It didn’t matter at all. These are extremely complicated concepts that 90% of people will never understand.
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