Question

Approximating the length of a curve: Suppose you want to

calculate the driving distance between New York City and

Dallas, Texas.
How accurate do you think your approximation is? Is it an over-approximation or an under-approximation?

Short answer

The driving distance found by the method of "subdivide, approximate and add up" is not so accurate.

Step-by-step solution

Step 1. Given Information

The major route between New York City and Dallas, Texas by driving distance.

Step 2. Analyzing the map 

The major route between New York City and Dallas, Texas is highlighted in the map above.
The approximate distances are:
Dallas-Lake Charles$345.3$miles
Lake Charles-New Orleans$205.4$miles
New Orleans-Panama City$329.2$miles
Panama City-Florida Gulf Coast$5.8$ miles
Florida Gulf Coast-Miami$141.7$ miles
Miami-Orlando$235.9$miles
Orlando-Savannah$280.5$miles
Savannah-Charlotte$251.9$miles
Charlotte-Washington DC $400.1$miles

Washington DC-New York $224.9$miles

Step 3. Comparing the distances

The total approximate distance is$\sum _{i=11}^{n-10}{x}_i$
$\sum _{i=11}^{n-10}{x}_i=345.3+205.4+329.2+5.8+141.7+235.9+280.5+251.9+400.1+224.9$
$\sum _{i=11}^{n-10}{x}_i=2420.7$ miles
The actual driving distance between New York City and Dallas, Texas is$1548.3$miles.
So, we can say that there is a huge difference between the actual distance by direct route when compared with the approximate distance found by "subdivide, approximate and add up".