# Big Picture: Integrals | Summary and Q&A

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May 5, 2010
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MIT OpenCourseWare
Big Picture: Integrals

## 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.

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### 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.