Scenario A
In the following scenario, we consider the distribution of a quantity along an axis.
Suppose that the function \(c(x) = 200 + 100 e^{-0.1x}\) models the density of traffic on a straight road, measured in cars per mile, where \(x\) is number of miles east of a major interchange, and consider the definite integral \(\int_0^2 (200 + 100 e^{-0.1x}) \, dx\text{.}\)
(i) What are the units on the product \(c(x) \cdot \triangle x\text{?}\)
Units:
cars
miles
cars/mile
miles/car
cars*cars
miles*miles
(ii) What are the units on the definite integral and its Riemann sum approximation given by
\begin{equation*}
\int_0^2 c(x) \, dx \approx \sum_{i=1}^n c(x_i) \triangle x?
\end{equation*}
Units:
cars
miles
cars/mile
miles/car
cars*cars
miles*miles
(iii) Evaluate the definite integral \(\int_0^2 c(x) \, dx = \int_0^2 (200 + 100 e^{-0.1x}) \, dx\) and write one sentence to explain the meaning of the value you find.
Scenario B
In the following scenario, we consider the distribution of a quantity along an axis.
On a 6 foot long shelf filled with books, the function \(B\) models the distribution of the weight of the books, measured in pounds per inch, where \(x\) is the number of inches from the left end of the bookshelf. Let \(B(x)\) be given by the rule \(\displaystyle B(x) = 0.5 + \frac{1}{(x+1)^2}\text{.}\)
(i) What are the units on the product \(B(x) \cdot \triangle x\text{?}\)
Units:
pounds
inches
pound/inch
inch/pound
pound*pound
inch*inch
(ii) What are the units on the definite integral and its Riemann sum approximation given by
\begin{equation*}
\int_{12}^{36} B(x) \, dx \approx \sum_{i=1}^n B(x_i) \triangle x?
\end{equation*}
Units:
pounds
inches
pound/inch
inch/pound
pound*pound
inch*inch
(iii) Evaluate the definite integral \(\displaystyle \int_{0}^{72} B(x) \, dx = \int_0^{72} (0.5 + \frac{1}{(x+1)^2}) \, dx\) and write one sentence to explain the meaning of the value you find.