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## Multivariable calculus

### Course: Multivariable calculus>Unit 4

Lesson 5: Double integrals

# Double integral 1

Introduction to the double integral. Created by Sal Khan.

## Want to join the conversation?

• I might be getting ahead of myself, but with this new definition do we use the same techniques (Integration by Parts, Trig. Substitution, etc) on Double Integrals as we did with single Definite Integrals? or do we need to redefine them? •  The beautiful thing is that all of the old techniques apply. A double integral is just one regular integral inside of another. Thus, you can use integration techniques on the inside integral, then use integration techniques for the outside integral.
• Would i have to do this i college if i did calculus. • That depends on the school. Often the first semester is differential calculus, the second is single integral calculus. After that, multivariable and vector calculus.
If you need the calculus for a non math degree, often the first two semesters are enough and often, the college will have a special calculus program for non-science students.

The question is: How far down the rabbit hole do you want to go?
The deeper you go, the more fun and amazing . . . . jump in!
• when in life would i use this?
(1 vote) • When you ask that question, you sort of prevent yourself from ever getting an answer. Learn it, see what it means, see how it works, and where it can be applied will reveal itself to you. You'll begin to see where others use it, and how such things can be used to describe our world, and how they can be used to try and make the world a better place.
• So this is a small question, probably irrelevant to the whole solving process...but when we labeled the surface f(x,y)dx as f(y),
how is f(y) the surface?...like how does f(y) represent the surface we integrate across y to find the volume? • f(x,y)dx isn't a surface, neither is f(y).

f(x,y)dx is the area of a small vertical planar element of the cross section of the volume under the surface f(x,y).

f(y) is the integral of f(x,y)dx so it's the sum of these areas and so gives the total area under the cross-section of the volume underneath the surface f(x,y) for a given value of y.

So, f(x,y) is a surface, but f(x,y)dx and f(y) refer to areas not on the surface, but of a section of the volume underneath it.

Following on further:

f(y)dy is the volume of a small section of the total volume under the surface f(x,y) whilst the integral of f(y)dy is the total volume of the volume under the surface f(x,y)
• Is this section just videos or are there problem sets attached to them? I'm using Khan Academy as a refresher for the GRE subject test but I can't seem to find problems after their Integral Calculus Section. • at Sal says "0 to b". why wouldn't it be a to b? • No because "a" is some constant y value and he is only evaluating the function with respect to changes in X from 0 - b (or some number more than 0 on the X axis). The inner integral, integrated from 0 to b provides the 2D value of area. Notice, however, that when a change of y is introduced, volume (a 3D value) can be calculated. In order to do that, there needs to be some change in y (0-a) by which the integral of 0 - b can be evaluated. Hope this mades sense... explaining calc with text is a bit tricky ;)
• Hello,
My questions concerns solving double integrals through conversion to polar coordinates. I was wondering why dxdy is sometimes replaced by dthetadr and other times by drdtheta? Does the order matter? • What?! No problems/questions?! I don't just want to watch a video. What fun is that?! • Whats the difference between double integrals and triple integrals?  