Cartoon Calculus: Ch 12 (Integration the easy way)

Quick links to Front matter, Back matter, and:
Part One: Ch 1: Introduction, Ch 2: Speed, Ch 3: Area, Ch 4: Fundamental theorem, Ch 5: Limits
Part Two: Ch 6: Derivatives, Ch 7: Toolkit, Ch 8: Extreme, Ch 9: Optimization, Ch 10: Economics
Part Three: Ch 11: Hard way, Ch 12: Easy way, Ch 13: Revisited, Ch 14: Physics, Ch 15: Conclusion

Page 147: A journey of a thousand miles…

WM: Is there a reason in Chapter 12 you don’t generalize to the integral power rule the way you generalized to the derivative power rule? You’re 3/4 of the way there just given the explanations you have already written. YB: We needed the generalized derivative power rule. We don’t need the generalized integral power rule.

Pages 148-149: In Chapter 4, we noted… // As we’ll see in this chapter

MB: p. 148 Again [as in Chapter 4, suggesting "inverses" instead of "opposites"], I think that instead of opposites, you want to say that you do something to f[x] (take its derivative) and you get f’[x]. Now what do you have to do to get back to f[x]? YB: Will ponder.

Pages 150-151: As we’ll see, finding anti-derivative… // To see how super easy it is…

MB: p. 150 Good and bad anti-derivatives—did you make up these terms? They are another distraction; at this level you don’t have to emphasize peculiar functions. YB: Yeah, I did make up these terms. But I’m not sure a better way to do it.

MB: p. 151 I suggest just integrating to a particular value of x, say x_0; not a constant (1). Then clearly F[x_0] is another function. To find its derivative, imagine finding F[x_0+h]. Draw the picture. Fancy that! The derivative of F[x] is just f[x]!

Pages 152-153: More generally, the easy way… // This result is so important…

Pages 154-155: The best way to find a good anti-derivative… // In this case, trial and error…

MB: p. 155 Don’t forget the ‘dx’ in integral. YB: Good suggestion. But will it get in the way of the graphic?

Pages 156-157: Unfortunately, there’s no foolproof method… // And every time you learn something new…

MB: p. 156 Why this function?

MB: p. 157 Misprint in first equation (x^(-2), not x^2). YB: Fixed in the latest draft, thank you!

WM: Page 157 – I don’t get the Chekhov’s gun reference. [And then later: I see the Chekhov's gun thing reappears in Chapter 14, so you do go back to it.]

Page 158: One final point…

Quick links to Front matter, Back matter, and:
Part One: Ch 1: Introduction, Ch 2: Speed, Ch 3: Area, Ch 4: Fundamental theorem, Ch 5: Limits
Part Two: Ch 6: Derivatives, Ch 7: Toolkit, Ch 8: Extreme, Ch 9: Optimization, Ch 10: Economics
Part Three: Ch 11: Hard way, Ch 12: Easy way, Ch 13: Revisited, Ch 14: Physics, Ch 15: Conclusion

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