### Homework

Estimate $\int_1^4 {1\over x} dx$ using $n = 6$ rectangles and
a. the right hand rule.
b. the left hand rule.
c. the average of a and b.

Solution. The width of each interval is $\Delta x = {{b-a}\over n}= {1\over 2}$ … so we have
a. ${1\over 2} ({2\over 3}+{2\over 4}+{2\over 5}+{2\over 6}+{2\over 7}+{2\over 8})= {{341}\over{280}}$.
b. ${1\over 2} ({2\over 2}+{2\over 3}+{2\over 4}+{2\over 5}+{2\over 6}+{2\over 7})= {{223}\over{140}}$.
c.${{787}\over{560}}$.
Note that the terms of each sum are the reciprocals of the endpoints $1 (={2\over 2}), {3\over 2}, ...,{8\over2}$; this is of course because our integrand ($1\over x$) is the reciprocal function. Finally, since the integral is “obviously” equal to $ln(4) \sim 1.386$, and ${{787}\over{560}} \sim 1.405$, our answer in c is reasonable.

#### 1 Comment

1. this is still way too much trouble
since, as far as i’ve been able to
discover, you can’t just TeX stuff
a page at a time.

• ## (Partial) Contents Page

Vlorbik On Math Ed ('07—'09)
(a good place to start!)