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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It’s the inverse operation of a derivative.
F'(x) = f(x)
∫f(x)dx = F(x)
It’s the amount of area under a given curve.
If we take a curve and break the area underneath it into several rectangles, the height of a given rectangle is f(x), and the width of the rectangle is Δx
The area of one of those rectangles would be f(x) * Δx
Now, we need to add all of those rectangles together.
If we split the area into n rectangles, and we integrate from a to b, so our rectangle heights are f(a), f(a+Δx), f(a+2Δx),… until f(b). The width of all the rectangles is Δx.
Δx = (b-a)/n
Now we just add them all together. Σ (i=0->n) f(a+i Δx)Δx
When we take the limit as n->infinity, then it becomes an integral
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