Practice problems for these concepts can be found at: Applications of Definite Integrals Practice Problems for AP Calculus

### Summary of Formulas

Position Function: *s* (*t*); *s* (*t*) = *v*(*t*) *dt*.

Velocity: *v*(*t*) = ; *v*(*t*) = *v*(*t*) *dt*.

Acceleration: *a*(*t*) = .

Speed: |*v*(*t*)|.

Displacement from *t*)_{1} to *t*)_{2} = *v*(*t*) *dt* = *s* (*t*_{2})–*s* (*t*_{1}).

Total Distance Traveled from *t*_{1} to *t*_{2} = |*v*(*t*)| *dt*.

### Example 1

The graph of the velocity function of a moving particle is shown in Figure 13.2-1. What is the total distance traveled by the particle during 0 ≤ *t* ≤ 12?

Total Distance Traveled

### Example 2

The velocity function of a moving particle on a coordinate line is *v*(*t*) = *t*^{2} +3*t* – 10 for 0 ≤ t ≤ 6. Find (a) the displacement by the particle during 0 ≤ *t* ≤ 6, and (b) the total distance traveled during 0 ≤ *t* ≤ 6.

The total distance traveled by the particle is or approximately 88.667.

### Example 3

The velocity function of a moving particle on a coordinate line is *v*(*t*) = *t*^{3} – 6*t*^{2} +11*t* – 6. Using a calculator, find (a) the displacement by the particle during 1 ≤ t ≤ 4, and (b) the total distance traveled by the particle during 1 ≤ t ≤ 4.

### Example 4

The acceleration function of a moving particle on a coordinate line is *a*(*t*) = – 4 and *v*_{0} =12 for 0 ≤ *t* ≤ 8. Find the total distance traveled by the particle during 0 ≤ *t* ≤ 8.

Total distance traveled by the particle is 68.

### Example 5

The velocity function of a moving particle on a coordinate line is *v*(*t*)=3 cos(2*t*) for 0 ≤ *t* ≤ 2π. Using a calculator:

- Determine when the particle is moving to the right.
- Determine when the particle stops.
- The total distance traveled by the particle during 0 ≤
*t*≤ 2π.

Solution:

- The particle is moving to the right when
*v*(*t*) > 0. Enter y 1= 3 cos(2*x*). Obtain*y*_{1}=0 when*t*= The particle is moving to the right when: - The particle stops when
*v*(*t*) = 0. - Total distance traveled |3 cos(2
*t*)|*dt*.

Thus the particle stops at *t* =

Enter (abs(3 cos(2*x* )), *x*, 0, 2π) and obtain 12. The total distance traveled by the particle is 12.

Practice problems for these concepts can be found at: Applications of Definite Integrals Practice Problems for AP Calculus

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