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 $\int_0^2 (200 + 100 e^{-0.1x}) \, dx$.

(i) What are the units on the product $c(x) \cdot \triangle x$?
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 $$\int_0^2 c(x) \, dx \approx \sum_{i=1}^n c(x_i) \triangle x?$$ 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.

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 $\displaystyle B(x) = 0.5 + \frac{1}{(x+1)^2}$.

(i) What are the units on the product $B(x) \cdot \triangle x$?
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 $$\int_{12}^{36} B(x) \, dx \approx \sum_{i=1}^n B(x_i) \triangle x?$$ 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.