A snail travels with velocity v(t) = 2 + 0.5sin t feet per second, where t is given in seconds. If the snail starts traveling at noon (t = 0), what does the expression
represent in the context of this problem?
When given an integral and asked to figure out what it means or what it represents, it's a good idea to first determine the units of the integral and the units of the variable of integration.
We're told that v(t) is measured in feet per second and t is measured in seconds. This means
is measured in feet. We're told that t = 0 corresponds to noon. Since t is measured in seconds, t = 60 seconds corresponds to 12:01pm.
Let's put all this information into a complete sentence. The integral
is the distance in feet the snail travels from 12:00 noon to 12:01 pm.
When you're told to explain what an integral is "in the context of the problem" or "in the language of the problem" or "in terms of the problem," we recommend first figuring out the units of the integral and the units of the variable of integration. The integral tells you how much a certain quantity has changed, and the limits of integration tell you the starting and ending points between which this change happens.