5 Steps to a 5 AP Calculus BC 2019

More Applications of Definite Integrals 313

13.2 Distance Traveled Problems

Summary of Formulas

Position Function:s(t);s(t)=

∫

v(t)dt.

Velocity:v(t)=

ds

dt

;v(t)=

∫

a(t)dt.

Acceleration:a(t)=

dv

dt

.

Speed:|v(t)|.

Displacement fromt 1 tot 2 =

∫t 2

t 1

v(t)dt=s(t 2 )−s(t 1 ).

Total Distance Traveled fromt 1 tot 2 =

∫t 2

t 1

∣∣

v(t)

∣∣

dt.

Example 1

See Figure 13.2-1.

0

10

–10

20

2 4 6 8 10 12

v(t)

v(t)

t

(seconds)

(feet/sec)

Figure 13.2-1

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=

∣∣

∣

∣

∫ 4

0

v(t)dt

∣∣

∣

∣+

∫ 12

4

v(t)dt

=

1

2

(4)(10)+

1

2

(8)(20)= 20 + 80 =100 feet.

Example 2

The velocity function of a moving particle on a coordinate line isv(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.