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Integral notations..

What does it mean for the integral to have a lower limit, but no upper limit?

Is there a difference between having [for example] the lower limit directly below the integral sign, rather than slightly in front?

I've seen multiple integral signs next to each other, and there is what appears to be a single lower limit for _all_ of them (this 'lower limit' is centered along the bottom, of these signs). Does this just mean the 'nth' integral of y, with a lower limit of 'whatever' ?

I've also seen integral signs with circles on their 'belly' (I think I made that term up... I'm sure you know what I mean though). What does this mean? What does it mean for a group of integral signs to be 'sharing' one of these circles ?
...I think I read something about it meaning the magnitude of the result -- so, the positive value.. or something.. ?

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3 Solutions
Harisha M GCommented:

1) Sometimes, upper limit is taken as infinity, or in some other cases, it denotes the other variable's value
2) No
3) No, it means the integral is to be carried for each variable given, in the order the differentials (dx, dy etc) appear
4) That is circular or contour integration, which means that the whole applicable values must be taken into account. It is generally used for area and volumes

Harisha M GCommented:
Probably, you are misunderstanding the meaning..

Sometimes, the double integral over x and y is written as below.. (which means integral over the area)

⌡⌡ f(x,y) . dA

and x, y and z as

⌡⌡⌡f(x,y,z) . dV

which are same as

⌡⌡ f(x,y) . dx dy
x y


⌡⌡⌡f(x,y,z) . dx dy dz
x y z

PS: Hope the characters get printed properly :)

Harisha M GCommented:
Harisha M GCommented:
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