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.*

**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. Label the axes. Show that your curve is y=x^2.
Then—the dx is NOT just added on for emphasis!!!!!*

**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.)
*

**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. Also, since Archimedes actually computed the area under x^2 and x^3, you should probably pull him in there somewhere.*

**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.*

*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.*

**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…**

**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