I’m in CalcII now and for the life of me cannot wrap my head around integrals. Now we are using things like u-Substitution methods and solving the areas between 2 curves. I can understand how the equations work, but not why because I still cannot picture what an integral is or why it’s important.
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The opposite of a derivative. Let’s say we’re going to make two graphs: graph 1 is my speed over time, graph 2 is my position over time.
For the whole time I am going at a speed of 1 (ft/sec or whatever). The first graph is a flat line at 1, because no matter what time it is, I’m going 10fps.
Graph 2 is the line y=x. At 0s I have gone 0ft. At 10s I have gone 10ft, etc.
Graph 1 is the derivative of graph 2, that is, the slope at any given point. Since the derivative of y=x is just y=1, that checks out. We can use it to find out how fast I’m going at any point (which is useful if my speed is variable, but in the case the answer is always 1 so meh).
Graph 2 is the integral of graph 1, which can be calculated as the area under the line. So how far have I gone in 10s using just graph 1? The area of the rectangle that is 1 (ft) high and 10s long is 10 (ft). This checks out because in graph 2, y=10 when x = 10 and in graph 1, the integral from 0 to 10 is 10.
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