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pull out your mobile phone for a second you have my phone there just scroll down just put on the desk and i just want you to look at it for a minute okay now human beings uh very inventive creatures okay um many creatures invent things but we're we're like invention is our theme right so we invent objects with the intent to generally solve a problem what problem does this is this design to solve uh people saying convenience convenience in what way like what kind of convenience is this soul emojis we invented this room i'm going to come back to that in a second um we invented phones uh because we wanted to talk to people i wanted to draw people from a distance right then we invented smartphones because we're like we want to be able to access the internet as well this can get into a network then then great now here's the thing emojis was a was a joke but actually it's not really we invent a thing with a purpose but when then we discover actually this thing is far more powerful than we originally intended and it can do lots of things that we didn't originally expect like trivially emojis or more seriously things like does anyone know what the arab spring is has anyone heard of that before the arab oh there's a nest there i'll use that the arab spring this was um this is about oh how many years ago was it now maybe six to seven years ago i think where um many arab nations sort of went into chaos because suddenly there were all these popular uprisings right people wanting to try to overthrow dictatorships and all that kind of thing and they came in a real way from these because suddenly people could communicate to each other in a secure way that the government could not control right so we invent things and we have an intent for them like find an area right a simple problem and then we discover they can do things these these inventions we have they can do things that we didn't necessarily expect so underneath this i want to make a little subheading which is an unexpected side effect okay now what i'm going to get us to think about is what's happened to my ipad there we go uh this function here okay now what we're going to do here is we're going to think about this this problem using integration and just see what happens out of just an invented problem okay so for instance if this is f of x 2 x minus 4 then what happens if we consider i've got some numbers here yeah from 0 to four what will happen okay we can treat this as an area or we can treat it as a primitive for now just trust me for a minute i'm going to treat it as a primitive just...