Integrating Multivariable Functions with Respect to One Variable  Calculus 3  Summary and Q&A
TL;DR
Integrating multivariable functions with respect to one variable involves adding an unknown function of the other variable.
Key Insights
 😀 In calculus 1, the plus C term is added to integrals of functions of one variable to represent the infinitely many possible solutions.
 When integrating a function of two variables with respect to one variable, an unknown function of the other variable must be added to maintain the partial derivative.
 When integrating a function of two variables with respect to the other variable, an unknown function of the first variable must be added to maintain the partial derivative.
 🪜 This concept of adding an unknown function of the other variable also applies to potential functions and solving exact equations in differential equations.
Transcript
hi everyone in this video I want to talk about integrating multivariable functions with respect to one variable so let's look at some examples so first let's go back to like calc one for a second let's say you have x squared DX so we have a function of one variable x squared and we're integrating with respect to X so you use the power rule so you a... Read More
Questions & Answers
Q: Why is the plus C added when integrating a function of one variable?
The plus C is added because the derivative of any constant is 0. Since the constant term can be any number, it must be included to represent the infinitely many possible solutions.
Q: How does integrating a function of two variables differ from integrating a function of one variable?
When integrating a function of two variables with respect to one variable, an unknown function of the other variable must be added. This is necessary to ensure that the partial derivative of the integrated function with respect to the variable being held constant results in the original function.
Q: What happens when integrating a function of two variables with respect to the other variable?
When integrating a function of two variables with respect to the other variable, an unknown function of the first variable must be added. This is done to ensure that the partial derivative of the integrated function with respect to the variable being held constant results in the original function.
Q: In what other areas of mathematics does this concept of adding an unknown function of the other variable arise?
This concept is important in potential functions and solving exact equations in differential equations. It is used to account for unknown functions that arise during the integration process.
Summary & Key Takeaways

In calculus 1, when integrating a function of one variable, the power rule is used, resulting in an additional constant term.

When integrating a function of two variables with respect to one variable, an unknown function of the other variable must be added.

The same rule applies when integrating with respect to the other variable, adding an unknown function of the first variable.