distance with integrals?

Question

Suppose a dog is running along a fence with velocity (in feet per second) v(t) = 10sin(t).
Using the fact that
∫π/2 sin(t) dt=1 and the interpretation of the integral in terms
0
of areas find in feet
a) the displacement of the dog after 57π/2 seconds
b) the total distance traveled by the dog after 57π/2 seconds

Answer 1

At t= 57π/2 the dog's displacement is 10 ft., and the total distance the dog has traveled is 570 ft.

Answer 2

recall that ∫v(t) dt = x(t) + x(0)

So integrating the velocity gives you the position

For part A, ∫10sin t dt from 0 to 57π/2
= -10cos t |0 to 57π/2 = -10feet

for part b you need to think about the fact that the dog is changing directions so you need to integrate from 0 to direction change then add the integral from direction change to the next direction change and so on. A graph might help.