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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An integral is like the sum function (capital sigma), but rather than being discrete (eg. Considers function values/graph heights at x-positions 1, 2, 3 and 4) it’s continuous (eg. considers all function values between x-positions 1 and 4.)
Think about a line on a graph, it can be any, say y = x^2
We can get an alright guess to the area under this curve by placing rectangles at even intervals underneath it, with some arbitrary (constant) width and being just tall enough to match the curve), eg:
Place 1-unit wide rectangles along the x-axis, and give them height such that each just about meet your graph (y=x^2). It’s not gonna look great, but you can find the area by hand easily (it’s discrete – there’s a countable number of rectangles). Now make them each half as wide (0.5-units wide), and fill the gaps with more bars. takes more to compute, but it’s a better approximation.
Do this forever, and if you generalise the process, you get integration! It’s called integration by first principles. Gives you an exact expression for the area under a curve.
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