Units and Integration

You may want to look at the other modules on Units.

Integration is a sophisticated form of multiplication. We often write the integral as a limit of Riemann sums.

Each term in a Riemann sum is a product. For example, suppose that the function f(t) gives the velocity of an object at time t and that the integral computes the change in the object's location from time t = a to time t = b. Suppose that velocity is measured in units of meters per second and that time is measured in seconds. Thus, the values of the function f(t) are measured in meters per second and the values of the variable t are measured in seconds.

The Riemann sums above estimate the change in position by first breaking the interval [a, b] into n subintervals of duration

Then the change in position during each interval is estimated by multiplying the duration of the interval, h, by velocity measured at a sample point in the interval. This product is measured in meters since

Thus, the Riemann sum gives us an estimate in meters.

Integral notation has several parts. In our example, the function f(t) is measured in units of meters per second. The symbol dt inside the integral sign can be thought of as representing the h appearing in the Riemann sum and, thus, is measured in seconds -- the same units as the variable t. The limits of integration, a and b, are also measured in seconds. The value of the integral is measured in meters.

The TI-92 screen below illustrates this

• The first line in this screen defines a constant g representing the acceleration of a free-falling object near the earth's surface. Because g represents acceleration, it is in units of meters per second².

• The second line defines a function v(t) representing the velocity of a free-falling object near the earth's surface at time t measured in seconds.

• Notice that when a value for t measured in seconds is put into the formula for v(t) the result is measured in units of meters per second. This is illustrated by the third line on the screen.

• The last line on the screen computes the integral -- that is, the change in the object's height -- from time t = 1 second to time t = 2 seconds. Notice that the result is measured in meters.