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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Derivatives lets you find the rate of something. Integrals let you accumulate the rate of something to to find the original thing.
Let’s start with what a derivative is since you are familiar with that. And let’s start simple.
For a linear line, you know the derivative is just the slope of the line. If y = 3/4x + 3, then y’ = 3/4. The derivative is the slope. And slope is just a rate. The change in y divided by your change in x. Or **rate** of how much y changes per change in x.
Now we go to a curve and we know how to find the derivative for that too. We know that as your change in x (or dx) becomes infinitely small, we get the instantaneous slope (or **rate**) of any line tangent to a point on the original curve.
Hopefully you can see where i’m going here. The derivative gives you the rate. In real world terms, your curve could describe the position of an object as it moves through space. Such position would be described in terms of feet or meters. The derivative with respect to *time* would give you the rate of position changing, measured as in units like feet per *second*, which we know as speed.
Now, let’s look at integrals. We’ll continue with our example of speed and position. Let’s pretend we have a straight highway between two cities. We know that if you know the speed, then you know how long it’ll take to go between the cities. We also know we can predict where you’ll be after a certain amount of time. This is simple algebra because our rate (speed) is constant and traveling is fairly intuitive to us because we experience it daily. But what if our rate is variable?
Well, pretend we have a curve describing our variable travel speed. Let’s keep it simple since we know how to do the math for a contant speed. Let’s roughly *estimate our average speed* every minute (dt), and calculate the distance traveled at that constant average speed. Or in other words, we are taking these snippets of **rates and accumulating** them all up. (You might recognize this as Riemann sums). This gives us the distance traveled and knowing the original position, the exact position.
Well now let’s treat it like we did with derivatives and lets make those time estimates tighter until our time interval is infinitely small. So now we are **accumulating those rates** instantaneously and instead of estimating position, we are precisely using rate (speed) to know position.
So if derivatives are finding the rate of change of something, integrals are using the rate of change of something, accumulating those rates to determine the value of the original thing.
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Now we can go expert level to recognize derivatives and integrals and shifts in dimension. We can go from position (feet), to speed (feet/second) to acceleration (feet/second^2) to rate of change in acceleration (known as jerk, feet/second^3) and back. All these being different orders of time, related to each other by integrals/derivatives of time.
In other words, acceleration can be thought of as the rate of speed changing. Or speed as an accumulation of acceleration. Jerk is appropriately how fast your acceleration changes (think car crash, where you rapidly decelerate).
We can also go from speed (ft/s) to flux (ft^2/s) to volumetric flow (ft^3/s) and back. Or just from a line, to an area, to a volume. Or any other number of dimensional translations and how they relate to each other. Some have tangible meaning, others are very abstract or meaningless, but remain mathematical constructs anyways.
Edit: aww fuck, realized some exponential formatting is off. RIP 3rd party apps that made this easier. I’m leaving it as a protest.
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