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 131: Part Three (introductory page)**

**Page 133: A journey of a thousand miles…**

*WM: I don’t have a lot of comments on Chapter 11. I think you did a perfectly good job, but this is a topic I think most Calc I students don’t really understand- they just live through it. You might tell readers they could skip this chapter if they like.* ~~ YB: Yeah, we kind of hint at that by calling this chapter “the hard way” and the next chapter “the easy way”. ~~

**Pages 134-135: In Part One, we learned… // The hard way is to directly climb Mt. Integral.**

*MB: p. 134 Explain that ‘area’ is not the number of square centimeters on the drawing, but area measured in units of x and y. * YB: Boy, I have no idea how to do this in a simple way.

* Label the axes. * ~~ YB: They are labeled. ~~

* Show that your curve is y=x^2. * YB: Good suggestion!

* MB: Then—the dx is NOT just added on for emphasis!!!!!* ~~ YB: I think this is okay as-is, and I do think the dx is for emphasis. (It would be more clear to have the limits of integration be x=0 to x=1, but part of the role of the dx is to demarcate the end of the integral. ~~

**Pages 136-137: The way to directly climb Mt. Integral… // The trick is to add more and more rectangles…**

*MB: p. 137 Give each rectangle a width Delta x and a height y[x] and remind readers that area of each rectangle is y Delta x. (Not all will remember this.) * ~~ YB: Agreed, but I think this will just confuse matters even more, especially since we haven’t previously used “Delta x”.~~

**Pages 138-139: Of course, there are many different paths up a mountain… // In the same way, there are…**

*RW: The disaster can be avoided by using Lebesgue integration and measure theory, so it might be funny if one of them was French. * YB: Good suggestion! Maybe we can hint at this with “Lesbegue” written on the cast or on the wheelchair or whatever is on the RHS of the frame?

* RW: Also, since Archimedes actually computed the area under x^2 and x^3, you should probably pull him in there somewhere.* ~~YB: No space for this I’m afraid. ~~

**Pages 140-141: As an example, let’s return… // This example turns out to be lucky…**

*MB: p. 140 Unless you explain (as in previous sentence) how you are calculating these areas, your formulae will be baffling.* YB: Yeah, this is hard. We’re trying to hint at that with the “width” and “height” labels, but that might not be enough. Maybe we can label the axes in the drawings, but that might make them too complicated. Worth a try though.

*MB: p. 141-146 This long discussion is a distraction and involves drawing some conclusions out of an unknown hat. These pages could be omitted and you can just make the point that adding up all those little rectangles is tedious and never ending.* ~~ YB: Spoken like a physicist! As a mathematician I actually like this stuff :)~~

**Pages 142-143: That example may not have seemed too hard… // That may seem impossible…**

**Pages 144-145: So let’s return to the same mountain… // Once again we’re lucky…**

*ME: couldn’t tell if you’re using ice cream sandwiches for the brain freeze. That would be cool (ha ha).* YB: Great idea! I’m not sure if we can do it graphically, but I will suggest to Grady.

**Page 146: That’s it!**

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