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I have a rudimentary understanding of integration as it applies to finding the area under a curve. I get the idea of adding up the areas of progressively smaller rectangles to approach the area, and that at an infinite number of rectangles the areas would be exactly the same. Right now I'm just playing around with the idea and I'm curious about how to sum up

For example, I've drawn a graph of f(x) = [tex]\sqrt{x}[/tex] between 0 and 1. Now this isn't like a geometric series where I can find the sum using [tex]S_{n}=a_{1}(1-r^{n})/1-r[/tex], because

Sorry for such a basic question, but how are things like this summed?

**n**number of things if the ratio between each one changes.For example, I've drawn a graph of f(x) = [tex]\sqrt{x}[/tex] between 0 and 1. Now this isn't like a geometric series where I can find the sum using [tex]S_{n}=a_{1}(1-r^{n})/1-r[/tex], because

**r**changes. I have discovered that [tex]a_{n} = a_{1}\sqrt{n}[/tex]. I have played around with the ratios and discovered some interesting patterns that emerge, and I have found a complicated way to sum up two objects, but it is really more work than just doing the sum directly, and there'd be no way to do it when [tex]n=\infty[/tex].Sorry for such a basic question, but how are things like this summed?

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