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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One simple way might be to think about it as the total result of a “function” which is always changing.
One example is an accelerating object. If an object is accelerating (a falling object is a basic case) then you can calculate the speed based on the time. However, that doesn’t tell you the distance it’s fallen, because the speed at each “instant” is different. You “kind of” add up the speed and time calculations for each minuscule instant of time, and you can get the distance covered. That same calculation done in one shot is an integral.
If you have a graph of speed on the y axis and time on the x axis, for an object that is accelerating, the graph will be a straight line (the speed is changing but doing so linearly), the “area under the curve” is a triangle shape, and the area of that triangle is the distance covered by the object.
If you graph the distance instead of the speed, distance on the y and time on the x, you get the positive side of a parabola. The distance graph is the integral of the speed graph.
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