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## AP®︎/College Calculus BC

### Course: AP®︎/College Calculus BC>Unit 6

Lesson 1: Exploring accumulations of change

# Introduction to integral calculus

AP.CALC:
CHA‑4 (EU)
,
CHA‑4.A (LO)
,
CHA‑4.A.1 (EK)
,
CHA‑4.A.2 (EK)
,
CHA‑4.A.3 (EK)
,
CHA‑4.A.4 (EK)
The basic idea of Integral calculus is finding the area under a curve. To find it exactly, we can divide the area into infinite rectangles of infinitely small width and sum their areas—calculus is great for working with infinite things! This idea is actually quite rich, and it's also tightly related to Differential calculus, as you will see in the upcoming videos.

## Want to join the conversation?

• Something I don't really understand but have been "pretending" to understand in class is what d really means. I understand where to put d/dx and dy/dx, but what does dx or dy really mean? I have been told that it is "an infinitely small change in" but then what does "in relation to" mean in the definition of dy/dx??
• Leibniz introduced the d/dx notation into calculus in 1684. The "d" comes from the first letter of the Latin word "differentia", and it represents an infinitely small change, as you said, or "infinitesimal". The Greek letter delta is also used to represent change, as in Δv/Δt, so dv/dt is not a big stretch.

The "in relation to" or "with respect to" that you refer to is the quantity in the denominator, and is normally the independent variable. If you take the derivative of a function with respect to x, that would be for a function of x, and is written as d/dx. For a function of time, as I wrote above, dv/dt would be the derivative of the velocity with respect to time, meaning that the function is written as a function of time. The velocity (the dependent variable) changes with respect to time (the independent variable), and it's derivative is acceleration.

Hope that helps.
https://en.wikipedia.org/wiki/Leibniz%27s_notation
• Okay, so integration is basically finding the area under a curve and it is kind of like the opposite of differentiation and hence is called the derivative. So does it mean that when you take the derivative, you are actually breaking up the curve into rectangular components?
• which class is appropriate to start learning calculus?
• Generally calculus (both differential and integral) is taught in junior and senior years (11th and 12th.) But there's absolutely no problem in learning it any time you want :)
I've learnt it during my freshman year (9th grade)!

I hope this helped!
• When Sal used the new notation at , I got confused. What does the notation "dx" mean in ∫ f(x) dx? Does it mean "with respect to x"? Or the derivative of something?

edit: Does dx in this case represent an infinitesimally small delta x?
• The "dx" indicates that we are integrating the function with respect to the "x" variable. In a function with multiple variables (such as x,y, and z), we can only integrate with respect to one variable and having "dx" or "dy" would show that we are integrating with respect to the "x" and "y" variables respectively.
• so when doing any sort of integral problems, am I essentially finding the area under the curve between the upper limit and the lower limit? or what else do we use integral for?

I feel really behind in class because I can't keep track of what to do when they give me a problem. sometimes I have to take antiderivative then plug the upper limit and the lower limit then subtract; and sometimes I just plug in the upper and lower limit then subtract.
• Yes, finding a definite integral can be thought of as finding the area under a curve (where area above the x-axis counts as positive, and area below the x-axis counts as negative).

Yes, a definite integral can be calculated by finding an anti-derivative, then plugging in the upper and lower limits and subtracting.
• hey , i have a doubt. what do you actually mean by d/dx(sin x )=cos x and integral of sin x = -cos x ....i have been trying to understand but i couldnt ..can you please explain this
• sin x(d/dx)= cos x

cos x(d/dx) = -sin x

-sin x(d/dx) = -cos x

-cos x(d/dx) = sin x

You can just accept the fact sin x(d/dx) = cos x and
cos x(d/dx) = -sin x or take a look at the proof theorem for sin(x) d/dx = cos x which is on this site.

Note that if you carry out the calculations in degrees for the proof you will get sin x(d/dx)= pi/180 cos x not cos x. Hence pi radians was defined to be equal to 180 degrees which simplified the equation.

Note: Sometimes there might be theorems with proofs outside the scope of high school however nothing will stop you from applying the theorem. So if you wish take a look at the proof. However, it is okay if you don't understand it so long as you understand the theorem. You can always come back to the proof a couple years in the future if you are interested.
• If dx becomes infinitely small, doesn't that mean that it moves closer and closer to zero and doesn't that mean that fx*dx will just approach zero?
• f(x) dx does approach zero, but the number of f(x) dx's approaches infinity. Adding together infinitely many infinitesimals generally gives you a finite value.
• When will we need to do antiderivatives vs definite integrals?
• Antiderivatives and integrals are the same exact thing.
• In general, how do you solve logarithmic or inverse tangent functions? Can anyone be clear about this?? I don't really understand. Also, is there two meanings to delta or does it just mean a change of quantity and so on? Wait I got one more question. if this is integral calculus, why not call it related to quantum calculus due to the calculation of infinitesimal objects?
• You can learn to solve logarithmic and inverse tangent functions by working through the lessons on that subject matter.
Delta (Δ) indicates the change in a variable.
Integral calculus has had that name for hundreds of years.
(1 vote)
• At he says f(x) sub i. Is i basically the number which the rectangle is. So like its the first rectangle to i=1? And should I do precalculus before this as I'm a student in yr9 and things in yr 9 and 10 are easy? Idk what the order is, Pre-Calculus, Calculus or Integral Calculus.
Thnx for the help
(1 vote)
• The order is Precalculus and then Calculus. Students often take APCalculus, because those courses are more geared to preparing students for the tests they must take, according to what I have read elsewhere on this site. Adults take Differential Calculus and then Integral Calculus, according to the sequence in textbooks I have. The adult courses contain everything from APCalculus, plus they may contain a bit more. I read that also elsewhere on this site. I'm not sure about Multivariable Calculus and Differential Equations, about which comes first, but I'm pretty sure they are after Differential Calculus and Integral Calculus. If anyone else can chime in on this, please feel free to do so.

## Video transcript

- [Instructor] So I have a curve here that represents y is equal to f of x, and there's a classic problem that mathematicians have long thought about. How do we find the area under this curve? Maybe under the curve and above the x-axis, and let's say between two boundaries. Let's say between x is equal to a and x is equal to b. So let me draw these boundaries right over here. That's our left boundary. This is our right boundary. And we want to think about this area right over here. Well, without calculus, you could actually get better and better approximations for it. How would you do it? Well, you could divide this section into a bunch of delta x's that go from a to b. They could be equal sections or not, but let's just say, for the sake of visualizations, I'm gonna draw roughly equal sections here. So that's the first. That's the second. This is the third. This is the fourth. This is the fifth. And then we have the sixth right over here. And so each of these, this is delta x, let's just call that delta x one. This is delta x two. This width right over here, this is delta x three, all the way to delta x n. I'll try to be general here. And so what we could do is, let's try to sum up the area of the rectangles defined here. And we could make the height, maybe we make the height based on the value of the function at the right bound. It doesn't have to be. It could be the value of the function someplace in this delta x. But that's one solution. We're gonna go into a lot more depth into it in future videos. And so we do that. And so now we have an approximation, where we could say, look, the area of each of these rectangles are going to be f of x sub i, where maybe x sub i is the right boundary, the way I've drawn it, times delta x i. That's each of these rectangles. And then we can sum them up, and that would give us an approximation for the area. But as long as we use a finite number, we might say, well, we can always get better by making our delta x's smaller and then by having more of these rectangles, or get to a situation here we're going from i is equal to one to i is equal to n. But what happens is delta x gets thinner and thinner and thinner, and n gets larger and larger and larger, as delta x gets infinitesimally small and then as n approaches infinity. And so you're probably sensing something, that maybe we could think about the limit as we could say as n approaches infinity or the limit as delta x becomes very, very, very, very small. And this notion of getting better and better approximations as we take the limit as n approaches infinity, this is the core idea of integral calculus. And it's called integral calculus because the central operation we use, the summing up of an infinite number of infinitesimally thin things is one way to visualize it, is the integral, that this is going to be the integral, in this case, from a to b. And we're gonna learn in a lot more depth, in this case, it is a definite integral of f of x, f of x, dx. But you can already see the parallels here. You can view the integral sign as like a sigma notation, as a summation sign, but instead of taking the sum of a discrete number of things you're taking the sum of an infinitely, an infinite number, infinitely thin things. Instead of delta x, you now have dx, infinitesimally small things. And this is a notion of an integral. So this right over here is an integral. Now what makes it interesting to calculus, it is using this notion of a limit, but what makes it even more powerful is it's connected to the notion of a derivative, which is one of these beautiful things in mathematics. As we will see in the fundamental theorem of calculus, that integration, the notion of an integral, is closely, tied closely to the notion of a derivative, in fact, the notion of an antiderivative. In differential calculus, we looked at the problem of, hey, if I have some function, I can take its derivative, and I can get the derivative of the function. Integral calculus, we're going to be doing a lot of, well, what if we start with the derivative, can we figure out through integration, can we figure out its antiderivative or the function whose derivative it is? As we will see, all of these are related. The idea of the area under a curve, the idea of a limit of summing an infinite number of infinitely things, thin things, and the notion of an antiderivative, they all come together in our journey in integral calculus.