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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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
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