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In order to replace the summation sign with the sign for an integral, one must focus on the object one wants to integrate, and it's environment. To simplify, one can say the the object one wants to integrate has a domain D1 in 2 or 3 space. The remainder, the environment, is all that we do not wish to integrate, which we can label D2. So thusly we create a rule saying that we will sum only over D1. This eliminates the environment, and will isolate the object. From this point, we then break down the object into sub-rectangles (most commonly in mathematics) and assign each subrectangle it's own set of coordinates. Thusly we can take coordinates from the lower right, upper right, lower left, and upper left corners of each subrectangle. Choose a system of orientation, if we choose lower-left, we will underestimate the summation, and if we choose upper right, we will over-estimate the summation. From this point we can say that the summation of the object is equal to the summation of all it's parts. A transative derrivation (property). So, we can say that the summation of D1 is therefore equal to the summation of each subrectangle and it's coordinates: R1[x, y] +R2[x, y] + ... etc.. An integral is the sum of parts over a defined area. So we can conclude that the summation of D1 is therefore equal to the integral of the subrectangles R within the domains of x and y according to the orientation of lower right corner or otherwise established. That's it in a nutshell, I suppose... lukeriverplate

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Q: How do you replace the summation sign with integration sign?
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What is the difference between summation and integration?

Integration is a special case of summation. Summation is the finite sum of multiple, fixed values. Integration is the limit of a summation as the number of elements approches infinity while a part of their respective value approaches zero.


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Integration uses a summation in the definition of the definite integral, so they are not the same, but they are related. They both yield a type of sum, or area (in the case of integration).


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It is not clear whether the question is 3a + 4a + 5a + 6a^3 which would be 12a + 6a^3 or 3a + 4a + 5a + (6a)^3 = 12a + 216a^3 or (3a + 4a + 5a + 6a)^3 = 5832*a^3 But none of these really require summation notation!


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Summation, represented by sigma (Σ) is the discreet version of integration. Integration is the continuous version of summation. It can be somewhat hard to explain the difference between discreet and continuous phenomena. The best way to think about integration is as the area under a line, curve, or function. Think of a triangle formed by lines y=0, x=1,and y=x. written mathematically, this is the integral x=0 to 1 of (x). You can also calculate the area of a half unit circle. integral x=-1 to 1 of (sqrt(1-x^2)). The best way to think about summation is the adding of numbers. sum x=0 to 1 of (x) yields the equation (0+1). sum x=-1 to 1 of (sqrt(1-x^2)) yields the equation (0+1+0).


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The mean is sometimes also known as the arithmetic average. For a finite number of observations, it is he sum of their values divided by the total number. It can also be described as the expected value of a variable. If a discrete numerical variable X can take the values x, then the mean is the sum [x*pr(X = x)] where the summation is over all possible values of x. For a continuous variable, replace the summation by integration.


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