Help Solve Math Term Paper Integration Error!

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The discussion centers around a mathematical error in calculating the distance traveled by an object with an initial linear velocity, v(0), that experiences opposing acceleration until it stops at time tf. The user correctly applies the Fundamental Theorem of Calculus but mistakenly concludes that the distance d(tf) equals zero, despite the object having traveled a distance before stopping. The correct approach involves integrating the velocity function over the time interval, rather than relying solely on the final velocity. The user seeks clarification on their misunderstanding regarding the integration process.

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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.
 
Last edited:
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MrMumbleX said:
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.
 
Last edited by a moderator:

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