Question 1

I think there's something wrong with the video. It says 7:53 minutes but it ends at 2:24 for me.

Accepted Answer

Hi Selim, this appears to be a issue on Khan Academy end. It seems like all the necessary content is covered in the video however.

I would suggest you move on to next part.

Question 2

what is dx suppose to represent in the notation?

Accepted Answer

Integration basically means summing up the area of infinitesimally thin rectangles under a function in a given interval [a, b]. The infinitesimally small width of the rectangles is denoted dx. Hope this helps!

Question 3

is this video supposed to be longer than 2:27? it feels like he got cut off. I didnt really learn anything from this because it feels like he didnt finish. I also see other people in the comments saying this but with no response.

Accepted Answer

Nope! This is how long this specific video is supposed to be. It continues later, but that's when you're supposed to learn more.

Question 4

Is there a way can do this process without graph?

Accepted Answer

At 1:36 in the video, Sal describes the mathematical notation for the definite integral of f(x) evaluated from a to b. There is no need for a graph at all, simply find the antiderivative of f(x), then evaluate that antiderivative at b, and subtract the antiderivative evaluated at a.

Question 5

Hi! I am a bit confused with what dx means in the video ... does anyone know?

Thanks!

Accepted Answer

dx,is always "a small change in x"  it amounts to the limit of x as x goes to 0.  or whatever is next tot he d

Question 6

what is the difference between exact area and definite integral?

Accepted Answer

The definite integral gives you a SIGNED area, meaning that areas above the x-axis are positive and areas below the x-axis are negative. That is why if you integrate y=sin(x) from 0 to 2Pi, the answer is 0. The area from 0 to Pi is positive and the area from Pi to 2Pi is negative -- they cancel each other out.

If you want to use integration to find a geometric (or exact, as you called it) area, you have to make all of the negative areas positive. One way to do this is with absolute values or just ignoring negative areas. For example, integrating y=|sin(x)| is not fun, so you normally break that into two integrals (0 to Pi and Pi to 2Pi) and ignore negative signs in the answers.

Question 7

Where is the video for finding the exact area using Riemann Sums?

Accepted Answer

(One little clarification: This is written assuming you already know what Riemann Sums are. If you don't, you can go check out the videos on that.)

It's nowhere, because there's no real way to do that. Riemann Sums are more of a way to get a sense of how much area is under the curve. We can talk about approximating with Riemann Sums, but actually evaluating them when the lines are infinitesimally small is a whole different story. There's a more precise (and most of the time easier) method called the Fundamental Theorem of Calculus that can tell you the exact answer. You can learn about that a bit later in this course. If using that doesn't work, we have computers that can approximate the area with Riemann Sums a lot more precisely than a person can, however, the answer will never be exact.

I'm a student, just like you, so I hope I know what I'm talking about, and that this response made any sense.

Question 8

Hello, I don't understand what is interesting about the area under the curve and between "a" and "b".

Accepted Answer

well, u can use it for more complex things. like later on u can find the volume of , say,a washer inside a cylinder by finding the area under the curves and all...

Course: AP®︎/College Calculus BC > Unit 6

Definite integrals intro

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