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Why bother with a post grad? Are they even worth it? Have your say! 26-10-2016
1. Hi, I'm currently reading the integration chapter in Bostock & Chandler The Core Course and in particular the part on "definite integration as summation" is slightly confusing me. Basically how they show that the area under a curve is an integral is by showing that . I understand the algebraic manipulation leading to the result; what I don't understand is what actually means.

For example, if I do the same thing for the gradient of a curve (instead of area) I get . Unlike , this result for gradient is understandable because it simply means that if you plug into the right hand side, it gives you a value of on the left hand side. For area however, plugging in into , we get on the left hand side, but what does this actually represent?!

I was thinking that because area is infinitesimal (unlike gradient, which is still defined even when you make ) you would need two values to find instead of just one like in differentiation, hence doesn't have a physical interpretation per se but does.

I hope you understand my confusion and any help would be appreciated.
2. (Original post by cloisters)

I hope you understand my confusion and any help would be appreciated.
Not entirely sure what's confusing you, sorry!

But: is defined as (i.e: summing up small rectangles underneath the curve and taking the width tending to 0). So are equidistant points spaced throughout and .

If your confusion is the indefinite integral, then it is just where the bottom limit is arbitrary, but it still comes down to area underneath a curve from a general starting point to a given point .
3. (Original post by Zacken)
Not entirely sure what's confusing you, sorry!

But: is defined as (i.e: summing up small rectangles underneath the curve and taking the width tending to 0). So are equidistant points spaced throughout and .

If your confusion is the indefinite integral, then it is just where the bottom limit is arbitrary, but it still comes down to area underneath a curve from a general starting point to a given point .
Hi, sorry I should have made myself clearer: the latter is what is confusing me, in other words what does an indefinite integral represent? I can see that is the area under the curve from to : what is ? I have tried to understand your post but I have some questions: if the lower bound is arbitrary for an indefinite integral then wouldn't (infinitely large) for say ?

Thanks
4. (Original post by cloisters)
Hi, sorry I should have made myself clearer: the latter is what is confusing me, in other words what does an indefinite integral represent? I can see that is the area under the curve from to : what is ? I have tried to understand your post but I have some questions: if it is the lower bound is arbitrary for an indefinite integral then wouldn't (infinitely large) for say ?

Thanks
Arbitrary just means that it doesn't matter. So if you want define , i.e: the integral from any point you want to a general point .

Not sure what you mean by the infinitely large bit, we're not quite working with improper integrals just yet, we're trying to wrap our heads around normal integrals, so the lower arbitrary bound has to be a real number (so not infinity) which means that will never be infinitely large.
5. (Original post by Zacken)
Arbitrary just means that it doesn't matter. So if you want define , i.e: the integral from any point you want to a general point .

Not sure what you mean by the infinitely large bit, we're not quite working with improper integrals just yet, we're trying to wrap our heads around normal integrals, so the lower arbitrary bound has to be a real number (so not infinity) which means that will never be infinitely large.
Sorry, just being honest here but I still don't get it: wouldn't ? I do sort of understand what you mean by being the area under the curve from an arbitrary point to but I'm completely failing to connect this to the fact that is a one-variable function.

And thanks for clearing that up that stuff! (Btw it says I can't rep you)
6. (Original post by cloisters)
Sorry, just being honest here but I still don't get it: wouldn't ? I do sort of understand what you mean by being the area under the curve from an arbitrary point to but I'm completely failing to connect this to the fact that is a one-variable function.
Yes, indeed! Which is precisely why when you integrate a function say you don't just get something involving , you also get an arbitrary constant - which appears precisely because of that arbitrary lower bound.

So, by the definition: , but the latter is just an arbitrary constant, so the same thing as .

Although it's a bit weird writing since is a constant. Kind of like referring to .
7. (Original post by Zacken)
Yes, indeed! Which is precisely why when you integrate a function say you don't just get something involving , you also get an arbitrary constant - which appears precisely because of that arbitrary lower bound.

So, by the definition: , but the latter is just an arbitrary constant, so the same thing as .

Although it's a bit weird writing since is a constant. Kind of like referring to .
Ah completely forgot about the constant - should've followed cgp's advice and got a +C stamp lol. But very interesting, I never thought of +C as being related to the indefinite integral like that although it makes complete sense in hindsight. Thank you very much!
8. (Original post by cloisters)
Ah completely forgot about the constant - should've followed cgp's advice and got a +C stamp lol. But very interesting, I never thought of +C as being related to the indefinite integral like that although it makes complete sense in hindsight. Thank you very much!
You're welcome!

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