The Definite Integral
Read this section to learn about the definite integral and its applications. Work through practice problems 1-6.
Units For the Definite Integral
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 is a measure of time in "seconds" and is
a velocity with units "feet/second", then has the units "seconds" and has the units
("feet/second")("seconds") = "feet," a measure of distance. Since each Riemann sum is a sum of
"feet" and the definite integral is the limit of the Riemann sums, the definite integral, has the same units, "feet".
If the units of are "square feet" and the units of are "feet", then is a number with the
units ("square feet").
("feet") = "cubic feet," a measure of volume. If is a force in grams, and is a
distance in centimeters, then is a number with the units "gram.
centimeters," a measure of work.
In general, the units for the definite integral are (units for ).(units for ). A quick check of
the units can help avoid errors in setting up an applied problem.