Average value, motion and accumulation in applied contexts

Learning objectives

Calculate the average value of a continuous function on an interval as (1/(b − a)) ∫ₐᵇ f(x) dx, and interpret it graphically and in context

Find the position of a particle from its velocity and an initial position using a definite integral, and the velocity from its acceleration and an initial velocity

Distinguish displacement, the integral of velocity, from total distance traveled, the integral of speed |v(t)|, and compute each from a formula, a table or a graph

Solve accumulation problems in which a quantity changes through a rate in and a rate out, finding the amount at a given time and the time at which the amount is greatest or least, with justification

Use the fundamental theorem of calculus to express the net change of a quantity as the integral of its rate of change, and interpret the result with the correct units