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### Course: Multivariable calculus>Unit 4

Lesson 5: Double integrals

# Double integrals 3

Let's integrate dy first! Created by Sal Khan.

## Want to join the conversation?

• Hi, could you help me please?

How do you invert the integration order? I cant find it in your videos.

Thanks!
• In the video "Double Integrals 2", Sal integrate first the x (considering "y" like a constant), and then substitute the values of the definite integral. Now, in this video, he integrate first the y (considering "x" like a constant). In the he mention that the result was the same, either you integrate first x or y.

I hope this helps.
• Why do you say Y to the third, instead of Y cubed or Y to the three, is it an 'American thing'? Im only wondering because it sounds very similar to Y to the 1/3 and could catch someone out.
• This is an interesting observation. As an American, I was always taught with the 'to the third' and never thought anything of it. I never realized the confusion that this could lead to!
• Does this work all the time? I mean can you always switch the way you integrate between dxdy and dydx and get the same answer? Also does this work for triple integrals as well?
• So does that make integration distributive?
• No, it just means that the order of integration does not matter, although sometimes, one order is much easier than another. Look up Fubini's Theorem for more on this.
• Hint: at it looks like sal write dy instead of dx. If you compare it with the other writings of the "y" you find out that he means dx -> to avoid confusion
• When a 3-D graph changes colors like in the pictures, does this denote anything, or is it just designed to better show depth?
• In this case, just to help visualize depth. You can think of it like 'shadows' or 'ray tracing light' on the plane in the graph.
In other cases that you may have seen (may see) with Engineering software that graphically depicts the application of forces on an object, you may see color changes used to graphically display the magnitude of the force (or heat being generated) being exerted across an area of the object.
As a real world example that you can try: Get or think of bending a piece of plastic back and forth (such as the tough plastic shells they package stuff in). As you bend it back and forth, the area around the bend will start to discolor and warp as it breaks down and eventually snaps or tears so you can get the object out of the packaging. That 'discoloration' would be the coloring of the graphic display along the bend to show the heat you are generating by exerting the force to bend the plastic.
• Is there any difference between the double integrals in this video and the double integrals that appear to be one single sign?
I have noticed on the program MathType that there is a symbol that looks like two integrals glued together, and they share the same boundary condition, and the program sems to treat it like one sign that just "looks" like two integrals.
But in this video Khan is essentially just stacking up several separate integrals instead.
(1 vote)
• They are equivalent.
A double integral is what you have seen on MathType, the two integral symbols together with a region definition below. By Fubini's theorem, we can break the double (or triple) integral up into an iterated or repeated integral, which is what you are seeing here on Khan and which is the norm.

Doing the integration is usually easy to do; more often it is the calculating of the boundaries of the region of integration in order to apply Fubini that is the most difficult part of solving a double or triple integral (well, that and possibly the ensuing algebra post integration) - but that is just my opinion.
• what do you do when you have to evaluate a double integral when no limits are given?..how do you chose which "slice" to take first and then give it a depth?ie with whose respect do we integrate first dx or dy? i hope i made sense while asking.. i couldn't explain my confusion in a concise manner
• Choose which ever makes evaluating the integral easier since by Fubini's Theorem, the order of integration does not matter.