Dy multiplied by y (order matters?)

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In summary, the conversation discusses an equation involving a variable y and the question of how to properly integrate it. It is concluded that the location of the differential dy can be changed without issue and the resulting integral will be of the form ∫y^2 dy.
  • #1
az91
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Hey all, this is my first post here,

I have a question that is kind of annoying me: I came across this equation:

dT=1/2 * [(m/L)dy]*[yx/L]^2. Only y is a variable here

I need to integrate it, but my question is do I change the location of the dy and solve it as: ∫y^2 dy? I believe that this is what is done in the book but I am not sure if this is correct.

Any help will be appreciated. :D
 
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  • #2
Yes, it becomes an integral of the form ##\int k\,y^2\,dy## for some constant ##k##, calculated from the parameters in the problem. ##dy## is considered as a differential and its location can be changed without problem.
 

1. What is the difference between "Dy multiplied by y" and "y multiplied by Dy"?

The main difference is the order in which the terms are multiplied. In "Dy multiplied by y", the derivative (Dy) is multiplied by the variable (y). In "y multiplied by Dy", the variable (y) is multiplied by the derivative (Dy). This can lead to different results depending on the specific values of y and Dy.

2. Does the order matter in multiplication when using "Dy" and "y"?

Yes, the order does matter. When multiplying "Dy" and "y", it is important to consider which term comes first. This can affect the final result and should be taken into account when solving equations or performing calculations.

3. Can "Dy multiplied by y" be simplified?

Yes, "Dy multiplied by y" can be simplified by applying the product rule of derivatives. This rule states that the derivative of a product is equal to the first term (Dy) multiplied by the derivative of the second term (y) plus the second term (y) multiplied by the derivative of the first term (Dy).

4. How is "Dy multiplied by y" used in calculus?

"Dy multiplied by y" is commonly used in calculus to find the derivative of a function. It is also used in solving differential equations and finding the rate of change of a variable with respect to another variable.

5. What are the applications of "Dy multiplied by y" in real life?

The applications of "Dy multiplied by y" in real life are numerous. It is used in fields such as physics, engineering, economics, and finance to model and analyze various systems and phenomena. For example, it can be used to calculate the velocity of an object, determine the growth rate of a population, or determine the rate of change of a stock price.

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