Dx in a function to be intigrated

• dE_logics
In summary, when integrating a function f(x), it is necessary to include dx in order to accurately calculate the area under the curve. However, x should not be replaced by dx, as it represents the small increment in x and is used in the integration process. This can be seen in the example of finding the area under the curve of f(x) = sin(x) / x from 0 to 5, where the integral is represented as \int_0^5 \frac{\sin{x}}{x}dx. By using dx, the resulting area is calculated with infinite accuracy.
dE_logics
Suppose in an integrated function f(x), its a necessity to have dx, why is it that all x in that function is not replaced by dx?...or is it?

Sorry but I really can't understand what you are trying to say? Could you illustrate it with some examples to show us what you mean?

I've seen this in many derivations.

Suppose the original function is (sin x)/x...and we're suppose to figure out the area.

So...the function to be integrated should be (sin dx)/dx...i.e replace all x with dx so they return results with infinite accuracy when the infinitely small sections are summed up.

But instead why do we take (sin dx)/x (I'm not sure if this is right, I've arbitrarily converted one of the x to dx).

You have it wrong, you don't replace any x's with dx's. Imagine moving your finger along the x-axis... x is the value of your finger is pointing to, f(x) is the value of the function at that point, and dx is a small increment in x. The small unit of area is thus f(x) dx, and you integrate f(x) dx.

Example: The area under the curve f(x) = sin(x) / x from 0 to 5 is
$$A=\int_0^5 \frac{\sin{x}}{x}dx$$

Humm...I see thanks!

$$\int_a^b f(x) dx$$ is an operator in itself, but think of as taking an infinitesimal length on the independent axis (this is dx), and multiplying it by the height of some function f(x) so you get the area under the curve and then summing it up from a to b.

Yep...as a result infinite accuracy.

1. What is the purpose of "Dx" in a function to be integrated?

"Dx" represents the infinitesimal change in the independent variable in the function being integrated. It is used to calculate the area under the curve of the function.

2. How is "Dx" calculated in a function to be integrated?

"Dx" is typically calculated using the limit definition of a derivative, where it is equal to the change in the independent variable divided by the change in the function's output.

3. Can "Dx" be positive or negative?

Yes, "Dx" can be either positive or negative, depending on the direction of change in the independent variable. When the function is increasing, "Dx" is positive, and when the function is decreasing, "Dx" is negative.

4. How does "Dx" affect the integration process?

"Dx" is a crucial component in calculating the area under the curve of a function. It allows us to break the function into infinitely small segments and calculate the area of each segment, then sum them together to find the total area under the curve.

5. Can "Dx" be changed or manipulated in the integration process?

"Dx" is a constant value in the integration process and cannot be changed or manipulated. It is an essential part of the mathematical definition of integration and is necessary for accurately calculating the area under the curve of a function.

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