# Integration: how to find Total Distance travelled?

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Particle P is initially at the origin. It moves with the velocity function v(t) = t2 - t - 2 cm/s.

a) Write a formula for the displacement function s(t).

• This is the integral of the velocity function. So, s(t) = t3/3 - t2/2 - 2t

b) Find the total distance travelled in the first 3 seconds of motion.

• Ok. This is the part I am having difficulty with.
• So, I set the v(t) = 0 and solved for t. I got t = 2 and t = -1
• I substituted those 2 numbers in the s(t) equation; s(2) = -10/3   s(-1) = -5/6
• I then did s(3) = 0 and s(0) = 0
• Doing that, the Total distance formula says to absolute vale the v(t) function. So I did  abs(v(3)-v(-1)) but didn't get the right answer. The answer correct is 5/6.

What am I doing wrong?

Edited by frank!e

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s(3) = 33 / 3 - 32/2 - 2(3) = 9 - 9/2 - 6 = 6/2 - 9/2 = -3/2

|s(2) -s(0)| + |s(3)-s(2)| = 10/3 + |-3/2 + 10/3| = 10/3 + (-9/6 + 20 / 6)  = 20/6 + 11/6 = 31 / 6 = 5 and 1/6 = 5.17 cm

As referenced here, on Wolfram|Alpha

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18 minutes ago, kw0573 said:

s(3) = 33 / 3 - 32/2 - 2(3) = 9 - 9/2 - 6 = 6/2 - 9/2 = -3/2

|s(2) -s(0)| + |s(3)-s(2)| = 10/3 + |-3/2 + 10/3| = 10/3 + (-9/6 + 20 / 6)  = 20/6 + 11/6 = 31 / 6 = 5 and 1/6 = 5.17 cm

As referenced here, on Wolfram|Alpha

Why did u not include when t = -1?

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Remember to look back at the original question, " Find the total distance travelled in the first 3 seconds of motion. " So since there is never a moment when t = -1, it's just from t = 0 to t = 3. (first 3 seconds)

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35 minutes ago, Elite X-Naut said:

Remember to look back at the original question, " Find the total distance travelled in the first 3 seconds of motion. " So since there is never a moment when t = -1, it's just from t = 0 to t = 3. (first 3 seconds)

That's understandable. SO, why can't we just have 3 = t2 and 0 = t1 in the formula? Do u know what I mean?

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Pure area is displacementwhich is the how far the ending position is from the starting position. It doesn't take account into whether the particle moved back and forth. Hence we need to find the zeros, and take absolute value of each interval where it is moving in the same direction, and add those up to find the total distance traveled. In this question from t = 0 to t = 2, the particle is moving in the negative direction.

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