My question is based around when you integrate dx as if it were a function and when it just represents the quantity that is infinitely small.(adsbygoogle = window.adsbygoogle || []).push({});

Say we have the function y=3, a straight horizontal line. We want to find the area under this line from x = 0 to x = 10. I know, you wouldn't use calculus to solve this because it is just a rectangle, but humor me. You could set up the integral like so:

[itex]\int[/itex]3dx with limits from 0 to 10. In this situation, 3 is a constant, so you can pull it out, leaving 3[itex]\int[/itex]dx from 0 to 10. Integration of the dx leaves just x, and plugging this into the limits gives a value of 10, which you then multiply by 3. This gives 30, the correct answer as easily calculated also by geometry.

Now we have the function y=x. We want to again find the area under the line from 0 to 10, so the integral becomes [itex]\int[/itex]xdx. Here when we integrate, we simply use the power rule to integrate, x becomes x^2/2, and we plug in our 10 value. We get 50, the correct answer, also easily calculated through geometry. The dx, effectively, was ignored during the integration process.

My question is: why in the first case did we actually use the dx and integrate it, but in the second example it was there just to show us what the differential is?

Thank you for your assitance.

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# Question on the functionality of dx

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