Day 32, Mat227

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Announcements:
- Your 6.8 homework: not done yet (graded 28, 66, 100)
- Your 7.1 homework is due next time.
- We have an exam through 7.1 on Friday next.
- Your 8.1 homework will be due Monday, 4/20, after the exam.
- We're going to jump to chapter 10 on Monday, so you might read ahead into 10.1.
Let's wrap up section 7.1: Integration by parts.
- First of all, the big picture: integration by parts is just the product rule backwards. This integration technique, like all integration techniques, is really just a differentiation technique in reverse.
  Consider the method of u-substitution, for example. That's just the chain rule in reverse.
- Derivation of the integration by parts formula:
  We start with the product rule:
  
  ${\left[u(x)v(x)\right]'=u(x)v'(x)+v(x)u'(x)}$
  Now integrate both sides:
  
  ${\int_{}^{}\left[u(x)v(x)\right]'dx=\int_{}^{}\left[{u(x)v'(x)+v(x)u'(x)}\right]dx}$
  
  ${\int_{}^{}u(x)v'(x)dx=u(x)v(x)-\int_{}^{}v(x)u'(x)dx}$
  
  ${\int_{}^{}udv=uv-\int_{}^{}vdu}$
- We'll motivate this technique by considering an important example: ${\int x\ln(x)dx}$
- Integration by parts may need to be carried out multiple times: sometimes the idea is to simplify the integral each time, until a really simple one arises allowing us to calculate the final solution (e.g. Example 6, p. 491). Sometimes it's something of a trick: we compute the integral multiple times in order to return to the original integral, allowing us to solve an equation for the original integral (e.g. Example 4, p. 490).
- Suggested strategy:
  - Choose u so that u' is simpler than u itself.
  - Choose v' so that $v=\int_{}^{}v'dx$ can be evaluated.
  - Sometimes v'=1 is a good choice.
- Of course, all this works with definite integrals:
- Examples:
  1. #9, p. 492 (sometimes the function to integrate is 1!)
  2. #10 (ditto)
  3. #26
  4. #39
  5. #40
  6. #51
  7. #66
Section 8.1: Arc Length
- What kinds of problems are we looking at in this section? Calculating the linear length of a curve (arc length).
- Let's consider arc length:
  
  ${L=\int{dL}\approx\sum_i\Delta{L_i}}$
  The usual trick is to discretize something continuous, and then pass to a limit:
  
  Now hopefully we can figure out how the formula will look:
  
  So what's the general formula?
- Examples:
  - #2, p. 567
  - #3, p. 567 (an Elliptic integral)
  - #37, p. 568

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