Calculus, difficult one!!!?

Question

Suppose a dog is running along a fence with velocity (in feet per second) v(t) = 8sin(t).
π/2
Using the fact that ∫ sin(t) dt=1 and the interpretation of the
0
integral in terms of areas find in feet
a) the displacement of the dog after 42π/2 seconds???
b) the total distance traveled by the dog after 42π/2 seconds??

Answer 1

The velocity is the rate of change of displacement with respect to time. So, the net change in displacement is simply the integral of velocity.

Assuming you start with a displacement of 0 at time 0, you can integrate from 0 to 42pi/2 of the velocity. x(t) = -8cos(t) + C. Taking that integral from 0 to 42pi/2, you plug in the numbers and get -8cos(42pi/2) - (-8cos(0)) = -8(cos(21pi) - cos(0)) = -8(-1 - 1) = 16. So, the answer to a is 16ft.

To do the total distance, you need to integrate over the absolute value. If you graph velocity, you'll notice that sin repeats. Taking the absolute value of that graph, you'll notice that there are lumps that repeat every interval of pi/2. Since the area under the graph is the integral, that's how you are to evaluate it. Using the given information, the area of one of the lumps is 1 (since the integral from 0 to pi/2 = 1). You have 42 of those lumps, so the total area is 42. That gives you a total distance of 42ft. Thus, your answer to part b is 42ft.

Answer 2

If
π/2
∫ sin(t) dt = 1
0
then
π/2
∫ 8sin(t) dt = 8
0
and
π
∫ 8sin(t) dt = 16
0
a) Because cos(t) is periodic, the displacement will repeat every 2π seconds.
42π/2 = 21π = 10(2π) + π= 10(2π) + 2(π/2), so the dog's displacement will be 16 ft.
b) The total distance traveled by the dog will be
8(42π/2)/(π/2) = 8*42 = 336

Answer 3

Do your own homework!

Answer 4

have you tried going to Math.com??? haha you got me there!!