Calculating Arc Length in Multivariate Calculus

In summary, the conversation discusses the problem of computing the length of a vector function, r(t), from t=0 to t=1. The formula for this involves taking the integral of the length of r'(t). However, the result of 5 keeps coming up, which is doubted by the participants. But upon further analysis, it is confirmed that the result is indeed correct, as r'(t) = 5 and represents the hypotenuse of a right triangle on a cylinder of radius 4.
  • #1
crazynut52
11
0
here is the problem, and I can't seem to get very far,

compute the length of r(t) = <3t, 4cost, 4sint> from t=0 to t=1

i know the formula is integral from 0 to 1 of length of r'(t)

but I keep coming up with 5, and it doesn't seem right, can someone please confirm or deny this. Thanks
 
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  • #2
that seems to be what i get too. r'(t) = 5, so when you integrate this from 0 to 1 it gives you 5 again.
 
  • #3
If you look at what you've plotted, i believe it is the hypotenuse of a right triangle with base 4 and height 3 wrapped around a cylinder [of radius 4].
 

Related to Calculating Arc Length in Multivariate Calculus

1. What is the definition of arc length in multivariate calculus?

Arc length in multivariate calculus is the length of a curve between two points. It is calculated by finding the integral of the square root of the sum of the squares of the derivative of each component of the curve.

2. How do you find the arc length of a curve in multivariate calculus?

To find the arc length of a curve in multivariate calculus, you must first parameterize the curve. Then, use the arc length formula, which involves finding the integral of the square root of the sum of the squares of the derivative of each component of the curve.

3. Can you explain the concept of arc length in terms of vectors?

Arc length can be thought of as the distance traveled by a particle along a curve, where the position of the particle is represented by a vector. The arc length of a curve is equal to the magnitude of the displacement vector between the initial and final positions of the particle.

4. Why is arc length important in multivariate calculus?

Arc length is important in multivariate calculus because it allows us to measure the length of a curve, which is essential in many real-world applications. It is also a fundamental concept in understanding the behavior of curves and surfaces in higher dimensions.

5. What are some common applications of arc length in multivariate calculus?

Arc length has various applications in fields such as physics, engineering, and computer graphics. It is used in determining the distance traveled by an object, the curvature of a path, and the design of curves and surfaces in computer graphics. It is also essential in optimization problems involving curves.

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