Reading: Accumulation in Real Life
We have already seen that the "area" under a graph can represent quantities whose units are not the usual geometric units of square meters or square feet. For example, if t is a measure of time in seconds and f(t) is a velocity with units feet/second, then the definite integral has units (feet/second) × (seconds) = feet. In general, the units for the definite integral [latex] \int_{a}^{b}f(x)dx [/latex] are (y-units) × (x-units). A quick check of the units can help avoid errors in setting up an applied problem. In previous examples, we looked at a function represented a rate of travel (miles per hour); in that case, the area represented the total distance traveled. For functions representing other rates such as the production of a factory (bicycles per day), or the flow of water in a river (gallons per minute) or traffic over a bridge (cars per minute), or the spread of a disease (newly sick people per week), the area will still represent the total amount of something.Example
Suppose MR(q) is the marginal revenue in dollars/item for selling q items. Then [latex] \int_{0}^{150}MR(q)dq [/latex] has units (dollars/item) × (items) = dollars, and represents the accumulated dollars for selling from 0 to 150 items. That is, [latex] \int_{0}^{150}MR(q)dq = TR(150) [/latex], the total revenue from selling 150 items.Example
Suppose r(t), in centimeters per year, represents how the diameter of a tree changes with time. Then [latex] \int_{T_1}^{T_2}r(t)dt [/latex] has units of (centimeters per year) × (years) = centimeters, and represents the accumulated growth of the tree’s diameter from T1 to T2. That is, [latex] \int_{T_1}^{T_2}r(t)dt [/latex] is the change in the diameter of the tree over this period of time.Licenses & Attributions
CC licensed content, Shared previously
- Business Calculus. Provided by: Washington State Colleges Authored by: Dale Hoffman and Shana Calaway. Located at: https://docs.google.com/file/d/0B1lkHWwO61QEM0gwOFhES2N5Tlk/edit. License: CC BY: Attribution.
