[12.1] - Reading Assignment

After reading sections 12.1

1. This question doesn't deal directly with integration but rather just asks to get acquainted (or re-acquainted) with the meaning of the double sum. Calculate the value of this double sum: (You should have six terms to add together...) $$\sum_{i=1}^2\sum_{j=1}^3 2^i3^j=?$$

   $$\begineq \sum_{i=1}^2\sum_{j=1}^3 2^i3^j&=&2^13^1+2^13^2+2^13^3+2^23^1+2^23^2+2^23^3\\ &=&6+18+54+12+36+108=234 \endeq$$

2. If we partition $[a, b]$ into $m$ subintervals of equal length and $[c, d]$ into $n$ subintervals of equal length, what is the area, $\Delta A$, for any subrectangle $R_{ij}$?

   All the sub-rectangles have the same area, $\Delta A$. So if we divide the total area $A=(b-a)(c-d)$ by the number of sub-rectangles, which is $m*n$ we get: $$\Delta A = \frac{A}{m*n}=\frac{(b-a)(c-d)}{m*n}=\frac{(b-a)}{m}\frac{(c-d)}{n}.$$
   

3. Muddy questions? Questions you wonder about?