What does area have to do with slope? | Chapter 9, Essence of calculus
Integrals are used to find the average of a continuous variable, and this can offer a perspective on why integrals and derivatives are inverses, distinct from the ...
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Here, I want to discuss one common type of problem where integration comes up: Finding the average of a continuous variable. This is a useful thing to know in its own right, but what’s really neat is that it gives a completely different perspective for why integrals and derivatives are inverses of one and other. Take a look at the graph of sin(x) between 0 and pi, which is half its period. What is the average height of this graph on that interval? It’s not a useless question. All sorts of cyclic phenomena in the world are modeled with sine waves: For example, the number of hours the sun is up per day as a function of which-day-of-the-year-it-is follows a sine wave pattern. So if you wanted to predict, say, the average effectiveness of solar panels in summer months vs. winter m...
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Here, I want to discuss one common type of problem where integration comes up: Finding the average of a continuous variable. This is a useful thing to know in its own right, but what’s really neat is that it gives a completely different perspective for why integrals and derivatives are inverses of one and other. Take a look at the graph of sin(x) between 0 and pi, which is half its period. What is the average height of this graph on that interval? It’s not a useless question. All sorts of cyclic phenomena in the world are modeled with sine waves: For example, the number of hours the sun is up per day as a function of which-day-of-the-year-it-is follows a sine wave pattern. So if you wanted to predict, say, the average effectiveness of solar panels in summer months vs. winter m...
