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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Calculus being what it is, it’s pretty hard to explain integrals using 5 year old terminology.
It’s the opposite of a derivative. A derivative tells us how fast a curve is moving up or down at a particular point. If you then graph the derivative, the area under that graph is the integral, and if you graph that area, you get a displaced version of the original function (moved up, or down, or in the same place)
Usually an integral tries to figure out how much space is under a graph. There are lots of rules or transforms to help figure out the answer when the graph gets complicated, such as the u substitution method you mention.
In physics we use integrals to figure out things like how much force a charge is applying over a large volume or area. It can also be used in engineering to figure out how much material is needed to fill or coat a structure with its shape defined by a function.
An integral is area under a curve. It is accumulated quantity. If you have an equation that defines how fast water enters a closed tank, the integral will give you the amount of water in the tank at the time specified, or the change in amount between two times.
Your exposure to integration should have started with a derivation of the Riemann integral as the limit of Riemann sums. This approach makes the concept and its geometric meaning very clear. Did your instructor skip this step?
One simple way might be to think about it as the total result of a “function” which is always changing.
One example is an accelerating object. If an object is accelerating (a falling object is a basic case) then you can calculate the speed based on the time. However, that doesn’t tell you the distance it’s fallen, because the speed at each “instant” is different. You “kind of” add up the speed and time calculations for each minuscule instant of time, and you can get the distance covered. That same calculation done in one shot is an integral.
If you have a graph of speed on the y axis and time on the x axis, for an object that is accelerating, the graph will be a straight line (the speed is changing but doing so linearly), the “area under the curve” is a triangle shape, and the area of that triangle is the distance covered by the object.
If you graph the distance instead of the speed, distance on the y and time on the x, you get the positive side of a parabola. The distance graph is the integral of the speed graph.
If you have a function that’s 1 everywhere (f(x)=1), then when you’re at 0 on the X axis, Y is already at 1, but the area is 0, because the shape you have under the graph has no width yet. when you’re at 1 on the x axis, the area under the part you have checked is 1 unit, because it draws a horizontal line from (0, 1) to (1, 1). Move to 2 on the x axis and the area will be 2 because you’re increasing it at a constant rate. The area under the function between x=0 and x=1 is a 1×1 square and you add a square to it in every step. The integral of f(x)=1 is f(x)=1x, because the area under the line increases at a linear rate, and that rate is 1.
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If you start with f(x)=2, then for every step on X, the area increases by 2, because instead of 1×1 squares, you get 1×2 rectangles. The integral of f(x)=2 is f(x)=2x, because for every step on the X axis, the area will be twice the value of X
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Now take a function whose Y value increases by 1 for every X value, like f(x)=2x. For every step on X, Y goes up by 2.
At the first step, the area will be 1, because at X=0, Y was also 0 and got up to 2 as a linear function, cutting the 1×2 rectangle into 2 triangles.
When you get to 2, you add the same triangle under your graph, but also a full rectangle that’s lined up with the previous triangle.
Your area was 1, now you added 3 to it, so it’s 4.
Next step, you start with 4, add a triangle and 2 rectangles to fill the area under the new triangle, which are 5 area units in total.
Add that new 5 to the existing 4 and you get 9.
If you look at the areas we got, they are 1, 4 and 9, which are 1², 2² and 3². The integral of f(x)=2x is f(x)=x²
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Finding the derivative function is the opposite of this. You look for how fast the function’s Y value “grows” at any point. The constant functions (f(x)=2) don’t change, don’t grow, so their derivative is f(x)=0. The linear function grows at a steady rate, the steeper it is, the highest the rate of that growth is. The derivative of f(x)=ax is f(x)=a, where ‘a’ is a constant number. The steepness, so the rate of growth of quadratic functions like x² grows at a steady pace, as we’ve seen in the integral example, so the derivative of x² is 2x
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.
The integral is the total amount of whatever you’re plotting. If you have a plot showing how fast a car is going over time, the integral of that plot will tell you how far the car went.
This is because it’s the area under the curve – you multiply the axes. In this case, it would be miles/hour on the Y axis and hours on the x axis. When you multiply you just get miles!
It’s essentially a sum, but only where you know the rule that defines a continuous set, rather than the discrete values.
With a regular sum, you might have 1 + 2 + 3 = 6
With an integral, you might know that the set follows a rule x=y and you can take an integral between two values of x to get the ‘sum’ of the continuous set between certain values.
The opposite of a derivative. Let’s say we’re going to make two graphs: graph 1 is my speed over time, graph 2 is my position over time.
For the whole time I am going at a speed of 1 (ft/sec or whatever). The first graph is a flat line at 1, because no matter what time it is, I’m going 10fps.
Graph 2 is the line y=x. At 0s I have gone 0ft. At 10s I have gone 10ft, etc.
Graph 1 is the derivative of graph 2, that is, the slope at any given point. Since the derivative of y=x is just y=1, that checks out. We can use it to find out how fast I’m going at any point (which is useful if my speed is variable, but in the case the answer is always 1 so meh).
Graph 2 is the integral of graph 1, which can be calculated as the area under the line. So how far have I gone in 10s using just graph 1? The area of the rectangle that is 1 (ft) high and 10s long is 10 (ft). This checks out because in graph 2, y=10 when x = 10 and in graph 1, the integral from 0 to 10 is 10.
One way to look at integration is of the inverse operation of differentiation. The derivative of x^2 is 2x; the integral of 2x gets you back to x^2.
The integral between two limits (values of the independent variable) gives you the area under the curve between those points. This can be useful for things such as calculating distance from a speed/time graph or for determining how much of a reactant is left in a chemical process after a certain length of time.
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