Distance Traveled by Point in Time Interval [1,3]

In summary, Olivermsun is telling you to rewrite an integral that is a sum of functions as the sum of the integrals of the functions.
  • #1
sarmen
5
0

Homework Statement


suppose the velocity of a point moving at time t, in seconds along a coordinate line is v(t)= (t+3)/(t^3+t) ft/sec. how far does the point travel during the time interval [1,3].

Homework Equations


The Attempt at a Solution



im not sure what to do, i used ∫3 (t+3)/(t^3+t) dt
this goes under the integral>> 1

but then i don't know what do next. please help
 
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  • #2
sarmen said:

Homework Statement


suppose the velocity of a point moving at time t, in seconds along a coordinate line is v(t)= (t+3)/(t^3+t) ft/sec. how far does the point travel during the time interval [1,3].

Homework Equations





The Attempt at a Solution



im not sure what to do, i used ∫3 (t+3)/(t^3+t) dt
this goes under the integral>> 1

but then i don't know what do next. please help

Rewrite (t + 3)/(t3 + t) using partial fractions decomposition. The denominator factors into t(t2 + 1), and the decomposition should look like A/t + (Bt + C)/(t2 + 1), for some constants A, B, and C.

Decomposition will give you two integrals to evaluate.
 
  • #3
You can try splitting the denominator into its components.
 
  • #4
ok so i did the partial fraction decomposition and got (3/t)+ (-3t+1)/ (t2 + 1).

do i dt it or just use the integrals [1,3] and plug it in?
 
  • #5
anyone?
 
  • #6
If
[tex]f(t) = g(t) + h(t)[/tex]
then
[tex]\int_a^b f(t) dt = ?[/tex]
 
  • #7
olivermsun said:
If
[tex]f(t) = g(t) + h(t)[/tex]
then
[tex]\int_a^b f(t) dt = ?[/tex]

dont we use the partial fraction that i got?
 
  • #8
Yes. olivermsun is telling you that you can rewrite an integral that is a sum of functions as the sum of the integrals of the functions. For your problem, you probably want to split it into three integrals.
 
  • #9
if i got (3/t)+ (-3t+1)/ (t2 + 1) then it would be

∫3/t dt + ∫(-3t/(t^2 + 1))dt +∫1/(t^2 + 1)dt

then I am not so sure
3ln|t|+ ? + tan^-1(x/1).

i don't know how to do ∫(-3t/(t^2 + 1))dt
 
  • #10
Consider the change of variables u = t² + 1.
 
  • #11
Norfonz said:
Consider the change of variables u = t² + 1.
IOW, an ordinary substitution.
 

1. What is meant by "Distance Traveled by Point in Time Interval [1,3]"?

"Distance Traveled by Point in Time Interval [1,3]" refers to the distance covered by an object or point during the specific time interval of 1 to 3 units of time. It is a measure of the total displacement of the object during that time period.

2. How is the distance traveled calculated during a specific time interval?

The distance traveled during a specific time interval can be calculated by finding the difference between the initial position of the object and its final position at the end of the time interval. This can be done using the formula: distance traveled = final position - initial position.

3. Can the distance traveled be negative?

Yes, the distance traveled can be negative. This occurs when the object moves in the opposite direction of its initial position during the time interval. For example, if an object starts at position 5 and ends at position 2 during the time interval [1,3], the distance traveled would be -3 units.

4. How is the distance traveled related to the speed of the object?

The distance traveled is directly proportional to the speed of the object. This means that the greater the speed of the object, the greater the distance traveled during a specific time interval. This relationship can be represented by the formula: distance traveled = speed x time.

5. Why is it important to measure the distance traveled by an object during a specific time interval?

Measuring the distance traveled by an object during a specific time interval is important for understanding the overall motion and displacement of the object. It can also help in calculating the speed and acceleration of the object, which are important factors in many scientific experiments and real-life scenarios.

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