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The last step is to replace hi with something more functional. And functional is the operative word, because the function determines the height of each rectangle. ” That is, for each hi in your sum (h1, h2, and so forth) you can replace the variable hi in the equation for an appropriate value of the function. Here’s how this looks: 25 26 Part I: Introduction to Integration This is the complete Riemann sum formula for the definite integral, so in a sense I’m done. But I still owe you a complete explanation for this last substitution, and here it comes.

You can verify this intuition by realizing that both 20 and 30 are less than the actual area, so whatever this area turns out to be, 30 must be closer to it. You might imagine that by slicing the area into more rectangles (say 10, or 100, or 1,000,000), you’d get progressively better estimates. 3333 . . In fact, you may very well decide to write: This, in fact, is the correct answer. But to justify this conclusion, you need a bit more rigor. Building a formula for finding area In the previous section, you calculate the areas of two rectangles and four rectangles, respectively, as follows: How high is up?

Y 4 y = 2x 3 2 1 Figure 2-1: Graph of the function y = 2x. ½ –2 –1 1 2 x In fact, assuming the continuity of the exponential curve even provides a rationale (or, I suppose, an irrationale) for calculating a number raised to an irrational exponent. This calculation is beyond the scope of this book, but it’s a problem in numerical analysis, a topic that I discuss briefly in Chapter 1. Noting trig notation Trigonometry is a big and important subject in Calculus II. I can’t cover everything you need to know about trig here.

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5 dimensional strictly locally homogeneous Riemannian manifolds by Patrangenaru V.

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