Big Picture: Integrals  Summary and Q&A
TL;DR
Calculus is about finding the relationship between two functions – the height (y of x) and the slope (s of x) – and understanding how to go from one to the other.
Questions & Answers
Q: What is the easiest way to go from function one to function two?
The easiest way is to recognize a formula for the slope and identify a corresponding height. For example, if the slope is x to the nth power, the height would be x to the n+1 power.
Q: What if the slope itself is x to the nth power?
In this case, the height function would involve an x to the n+1 power multiplied by (1/n+1), so that when the slope is taken, the n+1's cancel out and the power drops by one.
Q: Are there certain functions that can be recognized from a list to convert from function two to function one?
Some common functions that can be recognized from a list include x to the nth power, sine and cosine functions, and logarithmic functions. There are many other functions that do not fit into these categories, for which more complex methods may be required.
Q: What is the symbol for going from function two to function one?
The symbol is the integral symbol (∫), and the expression for going from function two to function one is the integral of the slope function with respect to x.
Summary & Key Takeaways

Calculus involves finding the relationship between the height and slope of a graph.

Going from function one (height) to function two (slope) involves taking small increments and finding the derivative.

Going from function two to function one involves recognizing patterns in the slope and integrating to find the height of the graph.