When do we integrate with respect to x or y?

Question:

When do we integrate with respect to x or y?

Anti-Derivatives:

Integrals, or anti-derivatives, are operations where we can find a function where our input is the slope or derivative. Integrals are widely used in many fields of science such as with area or volume determination, arc lengths, related parameters to acceleration such as velocity and position, among others.

Answer and Explanation:


The sign that we need to integrate with respect to x or to y is if we see either {eq}\displaystyle \rm dx {/eq} or {eq}\displaystyle \rm dy {/eq}, respectively. These differential elements, as they are so-called, allow us to do our integration with respect to the proper variable.

An integral is not valid if neither of these differential elements are present, and if they are not, our function might as well be a constant that does not need integrating. The differential elements are just as important as the integral symbol itself.

For instance, the integrals {eq}\displaystyle \rm \int\ x\ dx {/eq} and {eq}\displaystyle \rm \int\ x\ dy {/eq} will be very different, and we already know intuitively where to integrate because of our differential elements.


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Indefinite Integrals as Anti Derivatives

from Math 104: Calculus

Chapter 12 / Lesson 11
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