What is the Integral of x_2-x_1?

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In summary, the conversation discusses the use of Δ (delta) in a finite integral and an infinite integral, specifically in relation to the change in velocity of a raindrop as it descends. The final equation presented by the speaker is questioned and clarified in terms of the use of delta. It is concluded that a delta is not necessary in a finite integral as it is already accounted for, but is needed in an infinite integral to behave like a finite integral. The speaker provides a link to further explanation on the topic.
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Kuhan
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https://docs.google.com/file/d/0Bx0Sv7WqdghaOGR1YU9QUW5icWM/edit

Note: the partial differential equation (on the third line) is quoted from Erpul, G., Gabriels, D., Norton, L.D., 2003. The Combined Effect of Wind and Rain on Interrill Erosion Processes. Lecture given at the College on Soil Physics Trieste, 3 -21 March 2003 (LNS0418015):174-182. http://users.ictp.it/~pub_off/lectures/lns018/15Erpul.pdf [8 August 2012] and he makes a reference to "(Pedersen and Hasholt, 1995)" concerning the equation
 
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  • #2
You need to be clear in your mind about the physical process intended by the 'Δ'.
You expression for ΔVx is fine, if it means the change in horizontal velocity of a raindrop as it descends from z1 to z2. But then you introduce the notion of a delta in time as though it were independent. For a given raindrop, a delta in vertical height will be accompanied by a delta in time.
To arrive at your final equation, you seem to have shifted from Δx = ∫Vx.dt to Δx = ∫ΔVx.dt, which doesn't make any physical sense to me.
 
  • #3
Thanks! I managed to solve the problem.
I think the point I got wrong is that when we have a FINITE integral, it already counts as a 'delta' or 'change', thus we don't need to find the 'integral of a delta'

On the other hand, when we have an INFINITE integral, then we may 'integrate deltas'. Then, Δx=∫ΔV.dt if we use infinite integrals.

A clear example would be using graphs. The finite integral already shows us the area BETWEEN two curves or lines, so we don't have to find a 'delta' because the finite integral already gives the 'delta'.
Using a delta in an infinite integral basically makes the infinite integral behave like a finite integral without a delta.
 
  • #4
I've written an explanation (the answer) here : http://eraserboxtips.blogspot.com/2012/09/the-integral-of-delta.html
 
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FAQ: What is the Integral of x_2-x_1?

What is the Integral of x_2-x_1?

The Integral of x_2-x_1 is a mathematical concept that represents the area under the curve of the function x_2-x_1. It is denoted by ∫(x_2-x_1) and is used to find the total change in a quantity over a given range.

How is the Integral of x_2-x_1 calculated?

The Integral of x_2-x_1 is calculated by finding the anti-derivative of the function x_2-x_1. This can be done using various integration techniques such as substitution, integration by parts, or trigonometric substitution. Once the anti-derivative is found, the limits of integration are substituted into the equation and the resulting expression is evaluated.

What is the significance of the Integral of x_2-x_1?

The Integral of x_2-x_1 has many applications in physics, engineering, and economics. It can be used to find the displacement, velocity, or acceleration of an object, as well as the work done by a force. It is also used in calculating the area of irregular shapes and in determining the change in value of a variable over time.

Can the Integral of x_2-x_1 be negative?

Yes, the Integral of x_2-x_1 can be negative. This occurs when the function x_2-x_1 is below the x-axis, resulting in a negative area under the curve. It is important to pay attention to the limits of integration and the orientation of the function when evaluating the integral to avoid errors.

Are there any real-life examples of the Integral of x_2-x_1?

Yes, the Integral of x_2-x_1 has many real-life applications. For example, in physics, it can be used to calculate the work done by a variable force or the displacement of an object with varying velocity. In economics, it can be used to find the change in revenue over time. It is also commonly used in geometry to find the area under a curve or the volume of a three-dimensional shape.

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