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Well look at it this way. To find the exact area under a curve between two points (call them a and b), one can find the definite integral between the points. But we could also find it approximately.
Let a=(x1, y1) and b=(x2, y2). Let f be the function that the curve represents. So the point a=(x1,y1) on the curve represents the same thing as f(x1)=y1, that is applying function f to x1 to get result y1.
Now we can also say that at x1 the curve has height y1. This is good for our area problem because we know how that for a rectangle, area = height multiplied by width. A width will just be a difference in x value.
The horizontal distance between a and b is a width, = x2-x1. Can we just multiply the height by this width? Not really, because for any curve that isn't a straight horizontal line, the height will vary along that width, and so it isn't a rectangle.
But what if we divided up the area under the curve into thin vertical strips with equal width? Then these strips would almost be rectangles, so we could find the area of all of them individually and then add them to get a good approximation to the area. This is where the summation comes in.
Think of it another way. You have a curve graphed on a computer monitor. How would you calculate the area underneath it in pixels (dots on the computer screen)? Perhaps the most obvious way is to start at the first pixel of the curve and count how many pixels are underneath it, then do that for each pixel of the curve, adding up all the counts.
If we assume the curve is smooth, then the thinner we make the strips, the better the approximation is. Smooth means that if the more we zoom on a section of the curve, the more it appears to be a straight line. If it does appear that way then we can get the exact area by a triangle on top a rectangle. But we can ignore the area of the triangle because it will be vanishingly small because we have zoomed in so much.
In fact if we take it to the limit of an arbitrarily thin strip, we will get the integral and an exact answer to the area under the curve problem. It is said to be a continuous summation because it is summing the area in the way just described, and is continuous in that it is smooth, not chunky and blocky like with strips of definite thickness.
This concept of a limit of arbitrary smallness is the hardest concept to grasp in the calculus, but once you get it, you can understand all of the calculus with an ease you wouldn't have thought possible at first. Then it is just a matter of practice and memorizing to get good at it.
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What is difference between integration and summation?
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).
What is the difference between sigma and integral?
Sigma is a discrete sum, a sum with steps. Eg. add the numbers from 1 to 10 or add the numbers 1/2, 1/4,... A sigma always has a concept of a next thing to add, even if the list of things goes on forever. An integral is a continuous summation. It is a summation in that we are adding up the area under the curve, for example, but it is continuous in that because we are adding things of arbitrary smallness it's not really possible to always point to the individual terms that are being added because they become some kind of continuous blur. Instead we use some mathematical technique (integration). But still it is a kind of summation.
What does define mean in math terms?
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.
Why is a summation greater than an integral?
It's not. It depends on the method you use for summation whether summation > integral or integral > summation.
What is process of summation?
A summation is a recap of all the highlights of a presentation.
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.
Are summation and integration are same?
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).
What is difference between integration and summation?
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).
What is the primary function of wave summation?
produce smooth, continuous muscle contraction
What is the difference between sigma and integral?
Sigma is a discrete sum, a sum with steps. Eg. add the numbers from 1 to 10 or add the numbers 1/2, 1/4,... A sigma always has a concept of a next thing to add, even if the list of things goes on forever. An integral is a continuous summation. It is a summation in that we are adding up the area under the curve, for example, but it is continuous in that because we are adding things of arbitrary smallness it's not really possible to always point to the individual terms that are being added because they become some kind of continuous blur. Instead we use some mathematical technique (integration). But still it is a kind of summation.
What does continuous integration mean in the context of an iterative lifecycle?
Integration at the end of very iteration
What is difference bw integral and sigma?
summation is the discreet set of whole numbers whereas integration is the sum of all numbers.
What does define mean in math terms?
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.
What is difference between numerical integration and continuous system simulation?
No I don't know... Plz give me the right answer
Why is a summation greater than an integral?
It's not. It depends on the method you use for summation whether summation > integral or integral > summation.
Where does neural integration occurs?
Neural integration occurs mainly in the central nervous system, particularly in structures such as the brain and spinal cord. This process involves the summation and processing of incoming signals from various sensory receptors and other neurons to generate coordinated responses.
What has the author Adriaan C Zaanen written?
Adriaan C. Zaanen has written: 'Integration' -- subject(s): Generalized Integrals, Integrals, Generalized, Measure theory 'Continuity, integration, and Fourier theory' -- subject(s): Continuous Functions, Fourier series, Functions, Continuous, Numerical integration 'Introduction to operator theory in Riesz spaces' -- subject(s): Riesz spaces, Operator theory
