Calculus help please!!!!?

Question

An object moves along a line so that its velocity at time t is v(t)= 1/2 + sin2t feet per second. Find the displacement and the total distance traveled by the object for 0 less than or equal to t less than or equal to 3pi/2.

Displacement:_________ feet.

Total distance traveled: ________feet.

I have found the displacement to be 3.35619449019 but, I can not find the total distance traveled. Can someone please help me with this.... Thanks

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Answer 1

u can find the solution at the link
http://img133.imageshack.us/img133/1299/untitled302zh6.jpg

Answer 2

The velocity v at time t is given by:
v(t) = (1/2 + sin 2t) ft/s

Now, let d be the displacement/total distance (in this example the are equal). Therefore,
d'(t) = v(t)

Therefore,
d(t) = ∫v(t)

Therefore,
d(t) = (t/2 - cos 2t /2 + C) ft

Here, we must assume that at first the displacement/distance traveled is 0. Thus, the point (0,0) passes through the d function. Therefore, C = 0.
d(t) = (t/2 - cos 2t /2) ft

Therefore, since 0 ≤ t ≤ 3π/2, then the displacement is
from 0 ft to (3π/4 + 1/2) ft

^_^

Answer 3

Displacement will just be the integral from time 0 to 3pi/2.

Distance travelled take the absolute value of the function and then integrate it. One way to do this is to find where the function crosses zero, and then get the area (integral) between each of those points, make them all positive, and then add them.