- Length of curve to 4 decimal pl

In summary, the conversation is about finding the length of a curve using the formula $\displaystyle L = \int_c^d \sqrt{1+[g'(y)]^2}\, dy$ and how it differs from the formula $\displaystyle L = \int_{t_0}^{t_1} \sqrt{f^{2}(t)+ g^{2}(t)+ h^2(t)}dt$ when x is a function of y instead of y as a function of x. The person also mentions using a TI Nspire program to generate an image and the loss of their TI Inspire.
  • #1
karush
Gold Member
MHB
3,269
5
8.1.1
find the length of the curve to four decimal places
$y=xe^{-x},\quad 0 \le x \le 2$
eq from book
$L=\int_c^d\sqrt{1+[g'(y)]^2}\, dy$

ok I haven't done this in about 2 years and only did a few then so trying to review
rare stuff kinda

desmos graph
 
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  • #2
your cited book equation is for x as a function of y

the following is for the arclength of y as a function of x

$\displaystyle L = \int_0^2 \sqrt{1 + [e^{-x}(1-x)]^2} \, dx$

arclength1.jpg
 
  • #3
what generated the image?
 
  • #4
karush said:
what generated the image?

TI Nspire program on my laptop
 
  • #5
loaned out my TI Inspire
never got it back... :cry:
 
  • #6
If we are given that x= f(t), y= g(t), and z= h(t), for \[t_1> t> t_0\] then the arclength is \[\int_{t_0}^{t_1} \sqrt{f^{2}(t)+ g^{2}(t)+ h^2(t)}dt\].
 

1. What does "length of curve" mean?

The length of a curve refers to the distance along the curve from one point to another. It is measured by calculating the arc length of the curve, which takes into account the curvature of the curve.

2. Why is it important to measure the length of a curve?

Measuring the length of a curve is important in many fields of science, such as physics, engineering, and mathematics. It allows for accurate calculations and predictions, and can also provide insights into the behavior and properties of the curve.

3. How is the length of a curve calculated to 4 decimal places?

The length of a curve can be calculated using various methods, such as numerical integration or geometric formulas. To obtain a measurement to 4 decimal places, a high level of precision and accuracy is required in the calculation method and the input values.

4. What are some applications of measuring the length of a curve?

The length of a curve has many practical applications, such as in designing curved structures, analyzing the motion of objects along a curved path, and in modeling natural phenomena like ocean currents or planetary orbits.

5. Are there any limitations to measuring the length of a curve to 4 decimal places?

While measuring the length of a curve to 4 decimal places may provide a high level of precision, it is important to consider the limitations of the measuring method and the accuracy of the input values. In some cases, a higher level of precision may be required for more accurate results.

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