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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Calculus being what it is, it’s pretty hard to explain integrals using 5 year old terminology.
It’s the opposite of a derivative. A derivative tells us how fast a curve is moving up or down at a particular point. If you then graph the derivative, the area under that graph is the integral, and if you graph that area, you get a displaced version of the original function (moved up, or down, or in the same place)
Usually an integral tries to figure out how much space is under a graph. There are lots of rules or transforms to help figure out the answer when the graph gets complicated, such as the u substitution method you mention.
In physics we use integrals to figure out things like how much force a charge is applying over a large volume or area. It can also be used in engineering to figure out how much material is needed to fill or coat a structure with its shape defined by a function.
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