7.1 Integral as Net Change

Objective:SWBAT solve problems in which a rate is

integrated to find the net change over time in a variety of applications

Distance vs Displacement• The position of a function can be found by taking the integral

of the velocity function.• The change in position is a displacement (difference between

starting and ending points).• To see the difference between distance and displacement,

consider the following saying:“Two steps forward and one step back.”

• What is the total distance traveled?– 3 steps

• What is the total displacement?– 1 step forward

• Integrals will give you the total change in displacement.• To get total distance, you would need to incorporate

something like absolute value.

We had both positive (moved right) and negative (moved left) velocity, so what’s the change in position?

We are 4 spaces to the right further than what we started with.

Total distance means we want the total area.

Thinking about this as if we are taking steps, essentially Johnny Particle took 5 steps forward and 1 step back.

To find out how much sand is removed, just integrate the removal function from [0,6].

We have to some variable other than t since when we integrate we want the expression to be in terms of t (substitute in a t)

Remember that minimum values can occur at endpoints, or when the derivative is 0.