Finding the Length of y = x^{3/2} from x = 0 to x = 4

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In summary: We can use WolframAlpha to check your work, but as you see, we need to see your work first!Ohhh.BTW, Its a famous mistake. :)Yeah, lesson learned! I'll make sure to include my work next time. Thanks for the feedback!In summary, the length of the curve y = x^{3/2} from x = 0 to x = 4 is incorrect. The correct integral to evaluate is \int_0^4 \sqrt{ 1 + \frac{9}{4}x } \;\ dx, which can be solved using the substitution rule. It is important to change the limits of integration to be in terms of the new variable when using substitution.
  • #1
Precursor
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Homework Statement


Find the length of the curve. [tex]y = x^{3/2}[/tex] from x = 0 to x = 4.


Homework Equations


[tex]L = \int^{b}_{a} \sqrt{1 + (dy/dx)^{2}} dx[/tex]


The Attempt at a Solution


[tex]L = \int^{b}_{a} \sqrt{1 + (dy/dx)^{2}} dx[/tex]
[tex]L = \int^{4}_{0} \sqrt{1 + (3x^{1/2}/2)^{2}} dx[/tex]
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I used substitution rule
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[tex]L = 64/27[/tex]

Is this correct?

Thanks
 
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  • #2
Precursor said:

Homework Statement


Find the length of the curve. [tex]y = x^{3/2}[/tex] from x = 0 to x = 4.


Homework Equations


[tex]L = \int^{b}_{a} \sqrt{1 + (dy/dx)^{2}} dx[/tex]


The Attempt at a Solution


[tex]L = \int^{b}_{a} \sqrt{1 + (dy/dx)^{2}} dx[/tex]
[tex]L = \int^{4}_{0} \sqrt{1 + (3x^{1/2}/2)^{2}} dx[/tex]
.
.
.
.
I used substitution rule
.
.
.
[tex]L = 64/27[/tex]

Is this correct?

Thanks


http://www.wolframalpha.com/input/?i=integrate+sqrt[+1+++[+(3/2)+x^(1/2)+]^2+]+from+x=0+to+x=4
Its better to post all of your work.
 
  • #4
Precursor said:
So my answer is wrong afterall?

By the way, thanks a lot for that website! If only I knew such a website existed before. :smile:

Yeah Its wrong.

You will face: [tex]\int_0^4 \sqrt{ 1 + \frac{9}{4}x } \;\ dx[/tex]
What did you do for it?
 
  • #5
Actually I found [itex]c(7 \sqrt{7} - 1)[/itex],
where c is a numerical factor (I got 2/3) which I'm not sure of because I did the calculation sloppily.
So I suggest you show the rest of the calculation as well.

[edit] Too slow, you already have several replies. [/edit]
 
  • #6
Ratio Test =) said:
Yeah Its wrong.

You will face: [tex]\int_0^4 \sqrt{ 1 + \frac{9}{4}x } \;\ dx[/tex]
What did you do for it?

That's exactly what I got, but when I used the substitution rule, to integrate, I did not change the limits of integration to in terms of u.
 
  • #7
Precursor said:
That's exactly what I got, but when I used the substitution rule, to integrate, I did not change the limits of integration to in terms of u.

Ohhh.
BTW, Its a famous mistake. :)
 

FAQ: Finding the Length of y = x^{3/2} from x = 0 to x = 4

How do I find the length of the curve y = x^{3/2} from x = 0 to x = 4?

To find the length of the curve, we need to use the arc length formula: L = ∫[a,b] √(1 + (dy/dx)^2) dx. In this case, a = 0 and b = 4. We also need to find the derivative of y, which is dy/dx = (3/2)x^(1/2). Plugging these values into the formula, we get L = ∫[0,4] √(1 + (3/2)x^(1/2))^2 dx. Evaluating the integral, we get the length of the curve as approximately 7.26 units.

What is the significance of finding the length of a curve in mathematics?

Finding the length of a curve is important in mathematics because it helps us measure the distance between two points on a curve. This can have practical applications in various fields such as physics, engineering, and economics. It also helps us understand the behavior and properties of the curve.

Can I use a calculator to find the length of a curve?

Yes, you can use a calculator to find the length of a curve. However, depending on the complexity of the curve, the calculations may be tedious and time-consuming. It is recommended to use a graphing calculator or a computer program to find the length more accurately and efficiently.

Are there any other methods to find the length of a curve?

Yes, there are other methods to find the length of a curve, such as using the Pythagorean theorem and dividing the curve into small straight line segments. However, these methods may not be as accurate as using the arc length formula, especially for more complex curves.

What is the difference between arc length and chord length?

Arc length is the length of a curve between two points, while chord length is the straight line distance between two points on a curve. Arc length takes into account the curvature of the curve, while chord length does not. In most cases, arc length is longer than the chord length.

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