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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Many have posted the “Area under/between two wave forms” explanation, and yes. But — how does that work? This is how it was explained to me.
1. You have a wave (or two) and you can measure the Y distance at any X value.
2. Make a list of those values over a range. Consider those to be a bar graph with the bars having width of your X interval.
3. You can calculate the area of each bar (X * Y) and add them all up for a summary value. This would be (almost) the exact area between the two waves for any X to X+N. Except:
4. Unless both wave lines are parallel, there’s a crossing component at the end of each bar, making them trapezoids.
5. You could try again, calculating the areas of all those little upper and lower triangles, but it would still be a bit off because the wave lines are curved!
6. So — The Integral is the complete summation of all the vertical bars (infinitely narrow) including all the adjustments for the area of the little triangles (also infinitely small) at the ends.
Hope that helps!
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