Arc Length Problem Find 0 ≤ x ≤ 3

In summary, the arc length of y=x*e^(x^6) where 0 ≤ x ≤ 3 is approximately 3*e^(3^6), which is bounded above by the Manhattan distance and below by the length of the diagonal. The numerical integration on the bottom part of the curve can suggest an even more precise answer relative to y(3).
  • #1
bmr676
4
0

Homework Statement


Find the arc length of y=x*e^(x^6) where 0 ≤ x ≤ 3


Homework Equations





The Attempt at a Solution


I took the derivative of the equation and squared it to get (e^2(x^6))(1+12(x^6)+36(x^12)) then plugged it into the proper formula to get:
3
∫√(1+(e^2(x^6))(1+12(x^6)+36(x^12))) dx
0

I tried to plug it into my calculator but got the error message "Overload". I then tried Wolfram and got the answer 1.19685...*10^317, which seemed very extraneous.
One problem I thought about was maybe my area of integration was off, but couldn't think of another solution. Any suggestions or see any flaws?
 
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  • #2
Are you sure you copied the function correctly?

ehild
 
  • #3
absolutely sure...
 
  • #4
If it is so, you can not find the anti-derivative in closed form, and the arc length will be extremely great. It is certainly greater then the distance between the points (1,y(1)) and (3,y(3)) which is about 4˙10^316. Was not it x(e^x)^6 instead?

ehild
 
Last edited:
  • #5
The error is lost in the noise... the answer to a high degree of precision is 3*e^(3^6).

Rationale: it is a monotonic function, so the answer is bounded below by the length of the diagonal and above by the Manhattan distance. Both of those are also approx the above number.
 
  • #6
Numerical integration on the bottom part of the curve should suggest exactly how much greater than y(3) the arc length is. Hint: it's less than 1.
 
  • #7
Then the answer I stated above isn't as out there as I thought it might have been?
 
  • #8
The answer Wolfram gave is numerically correct.

It's more interesting to find the answer relative to y(3), since for practical purposes y(3) is the answer anyway.
 

1. What is the definition of arc length?

Arc length is the distance along the curved line of an arc, measured in linear units. It is the measure of the actual length of a section of a curve or circle.

2. How is arc length calculated?

Arc length can be calculated using the formula L = rθ, where L is the arc length, r is the radius, and θ is the central angle in radians. Alternatively, it can also be calculated using the formula L = 2πr(n/360), where n is the central angle in degrees.

3. What is the significance of the 0 ≤ x ≤ 3 in the "Arc Length Problem"?

The inequality 0 ≤ x ≤ 3 represents the range of values for the variable x in the given problem. In this case, it means that the arc length is being calculated for a curve or circle with a radius of 3 units, and the arc length is being measured between the starting point (x = 0) and the end point (x = 3).

4. Can arc length be negative?

No, arc length cannot be negative. It is always a positive value, as it represents the distance along a curve or circle.

5. How can the "Arc Length Problem" be applied in real life?

The "Arc Length Problem" can be applied in various fields such as engineering, architecture, and physics. For example, it can be used to calculate the length of a curved road or track, the circumference of a circular object, or the distance travelled by a rotating object. It is also used in computer graphics to create curved lines and shapes.

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