Integration Error?

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The following is from my investigation of a problem for my math term paper.
An object is given a certain linear velocity, v(0), has acceleration opposing its velocity, and the object comes to a stop at time = tf due to the acceleration.
d(tf) = integral from 0 to tf of v(t)dt = integral from 0 to tf of [v(0) + integral from 0 to tf of a(t)dt]dt, d(t) is distance function of time t, v(t) is velocity function of time, and a(t) is acceleration function of time.
By Fundamental Theorem of Calculus, integral from 0 to tf of a(t)dt = antiderivative of a(tf) -antiderivative of a(0)
-> Antiderivative of a(t) = v(t)
-> integral from 0 to tf of a(t)dt = v(tf) - v(0)
-> v(tf) = 0 since the cube comes to stop
-> = v(tf) - v(0) = -v(0)

So d(tf) = integral from 0 to tf of [v(0) + -v(0)]dt = 0 -> BUT d ≠ 0
BUT d = integral from 0 to tf of v(t)dt = integral from 0 to tf of [v(0) + integral from 0 to tf of a(t)dt]dt is TRUE, and integral from 0 to tf of a(t)dt = v(tf) - v(0) = -v(0) is also TRUE.

So what did I do wrong? I was thinking to take the absolute value of the integral of acceleration, but I don't see how that would make any sense.

Help PLEASE! Any thoughts and comments are also appreciated.
 
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HallsofIvy
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The following is from my investigation of a problem for my math term paper.
An object is given a certain linear velocity, v(0), has acceleration opposing its velocity, and the object comes to a stop at time = tf due to the acceleration.
d(tf) = integral from 0 to tf of v(t)dt = integral from 0 to tf of [v(0) + integral from 0 to tf of a(t)dt]dt, d(t) is distance function of time t, v(t) is velocity function of time, and a(t) is acceleration function of time.
By Fundamental Theorem of Calculus, integral from 0 to tf of a(t)dt = antiderivative of a(tf) -antiderivative of a(0)
-> Antiderivative of a(t) = v(t)
-> integral from 0 to tf of a(t)dt = v(tf) - v(0)
-> v(tf) = 0 since the cube comes to stop
-> = v(tf) - v(0) = -v(0)
Okay, the change in velocity is -v(0).

So d(tf) = integral from 0 to tf of [v(0) + -v(0)]dt = 0 -> BUT d ≠ 0
No. you integrate the velocity function from 0 to tf, Not just the final velocity which is what you are doing here.

BUT d = integral from 0 to tf of v(t)dt = integral from 0 to tf of [v(0) + integral from 0 to tf of a(t)dt]dt is TRUE, and integral from 0 to tf of a(t)dt = v(tf) - v(0) = -v(0) is also TRUE.

So what did I do wrong? I was thinking to take the absolute value of the integral of acceleration, but I don't see how that would make any sense.

Help PLEASE! Any thoughts and comments are also appreciated.
 
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