Calculus 3 Lecture 14.2: How to Solve Double/Repeated/Iterated Integrals

Calculus 3 Lecture 14.2: How to Solve Double/Repeated/Iterated Integrals: The techniques of solving Double Integrals with a focus on how to construct a double integral over a given region.

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[tauceti]

Context: Integrating under regions. What is R, how to find R, and how to evaluate when not given R (how to find)
2:06 check f(x, y) is a surface.
3:10 doesn't matter which order you find under the region dx, or dy, still yields same area.
where R is the Rectangular Region, which remember is based on the change in X and Y which both have intervals. Thus.
a<x<b | c<y<d
8:35 R corresponds the limits of integration whereas f(x, y) represents the function of the surface. You graph R to find the limits if also a function i.e. 1+1/x
9:32 make sure you match dx dy values with limit values of x and y
13:00 This dx=dy situation "Fubini's Theorem"
22:10 can check work by taking the partial of what you just did in the integral with respect to that variable
32:50 | 34:20 | 35:40 - Integrate between regions of the same 2 function. By doing dy inbetween x-function and axis. At x, your good at the bottom, then at the top its just top curled so u need another integral. Don't do dx in that case, start dy first. Don't let a purple line ruin your day. Hahahaha.
Remember A double integral gives volume
49:32 Cheat Codes! Between Constants Last. So if not 0 1<y<3 last... Notation -> dxdy.. What if x still doesnt complete the path, you need two integrals still then? You have to draw it out! Find the intercepts? WHAT INTERCEPTS wt are those. Where do the functions agree on values. Plug values in x/y. Find where they = 0. How to integrate from there. Locate your functions and your bounds. You'll have the constants a<x<b and the function for x. Also you can swap the function given. Say its given as y=x+4 -> x=y-4 then you have bounds for dx.
1:02:08

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