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### Course: Multivariable calculus>Unit 4

Lesson 5: Double integrals

# Double integrals 4

Another way to conceptualize the double integral. Created by Sal Khan.

## Want to join the conversation?

• Where do I get these almonds for power?
• some things should be kept secret
• Does Sal have any videos that integrate using polar coordinates?
• I agree, he should definitely make one using polar coordinates. I'm having trouble figuring out the limits of integration for theta.
• At , Sal mentions that dA is used as a shorthand. Isn't it also used to generalize dxdy so that the order of integration (and the coordinate system) is not specified? I feel like this would make it more applicable in proofs and theorems and such.
• dA is often used to indicate integration over an area without specifying how the integration will be performed. dA can even indicate integration over a curved surface in 3-D.
• In-spite of the approach to the problem being different from the first video. The way Sal goes about solving it is exactly the same. Is it only a way to visualise the problem differently? (since the volume of a column is different from a sheet etc.)
• Pretty much. In the first video it was cross sections of sheets intersecting, but in this video it's going strait to the matter of taking infinitely many columns of the space under the graph. They both get you the end, but take a slightly different point of view.
• i can`t understand what is the difference between simple integration and double integration,,in simple one we used to find area under the curve and in this(double integration) volume under the curve,,, so please explain me the difference b/w them,,maybe i am missing something
• I don't think you are missing anything at this stage. The only comment I have is that, yes, single integration is the area under a curve, whereas the double is volume under a surface (not curve as you wrote) if you are integrating a function such that z=f(x, y) (Remember - there is no volume under a curve - why?).

Is there something in particular you think should be different?

When you get to triple integrals, there is also a volume relationship, but there is also much more depending on the function being integrated - so the interpretation of the result of triple integrals can be a bit more complicated.
• Question: Do I always need to draw the two dimensional graph of the bounds? Or can I jump into integrating? Do I NEED to do this for the limits of integration? I'm not opposed to it, I Just prefer a mathematical way so that I don't have to think about how to draw it or take the time to do it on my calculator. Do I just solve for things in terms of x when I do dydx, and vice versa? And my teacher said you can't always just choose which order you want to do it. How do I know when I can and when I can't? Oh, and can you do one on polar coordinates? I'm having trouble finding the limits of integration for theta =/
• what is double integration of cos(x+y)dx.dy first integrating from pi/2 to x and second integration from 0 to pi/2 ?
• Hint: start by using the trig identity `cos(x+y) = cos(x)cos(y) - sin(x)sin(y)` ... I think you will find that this problem is relatively straightforward after that. HTH
• So what is the diff of double and triple integrals?
• I didn't knew in America, Europe education is this clear concept. I wish I would have done bachelor's degree in USA.

CAn you make video on double integral in polar just like the first method ? So we could visualize ??