5 Steps to a 5 AP Calculus AB 2019 - William Ma

MA 3972-MA-Book April 11, 2018 16:5

More Applications of Definite Integrals 321

Total Distance Traveled

∫ 2

1

v(t)dt+

∫ 3

2

−v(t)dt+

∫ 4

3

v(t)dt.

Enter

∫

(y1(x),x,1,2)and obtain

1

4

.

Enter

∫

(−y1(x),x,2,3)and obtain

1

4

.

Enter

∫

(y1(x),x,3,4)and obtain

9

4

.

Thus, the total distance traveled is

(

1

4

+

1

4

+

9

4

)

=

11

4

.

Example 4

The acceleration function of a moving particle on a coordinate line isa(t)=−4 andv 0 =
12 for 0≤t≤8. Find the total distance traveled by the particle during 0≤t≤8.

a(t)=− 4

v(t)=

∫

a(t)dt=

∫

− 4 dt=− 4 t+C

Sincev 0 = 12 ⇒−4(0)+C=12 orC= 12.

Thus,v(t)=− 4 t+ 12.

Total Distance Traveled=

∫ 8

0

∣∣

− 4 t+ 12

∣∣

dt.

Let− 4 t+ 12 = 0 ⇒t= 3.

|− 4 t+ 12 |=

{

− 4 t+ 12 if 0≤t≤ 3

−(− 4 t+12) ift> 3

∫ 8

0

∣∣

− 4 t+ 12

∣∣

dt=

∫ 3

0

∣∣

− 4 t+ 12

∣∣

dt+

∫ 8

3

−(− 4 t+12)dt

=

[

− 12 t^2 + 12 t

] 3

0 +

[

2 t^2 + 12 t

] 8

3

= 18 + 50 = 68.

Thus, the total distance traveled by the particle is 68.

Example 5

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

(a) Determine when the particle is moving to the right.

(b) Determine when the particle stops.

(c) Determine the total distance traveled by the particle during 0≤t≤ 2 π.