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 ∫abf(x)dx 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 ∫0150MR(q)dq has units (dollars/item) × (items) = dollars, and represents the accumulated dollars for selling from 0 to 150 items. That is, ∫0150MR(q)dq=TR(150), 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 ∫T1T2r(t)dt has units of (centimeters per year) × (years) = centimeters, and represents the accumulated growth of the tree’s diameter from T1 to T2. That is, ∫T1T2r(t)dt is the change in the diameter of the tree over this period of time.
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Business Calculus.Provided by: Washington State CollegesAuthored by: Dale Hoffman and Shana Calaway.Located at: https://docs.google.com/file/d/0B1lkHWwO61QEM0gwOFhES2N5Tlk/edit.License: CC BY: Attribution.