Find the length of the specified arc of the given curve

In summary, To find the length of the specified arc of the given curve, use the formula integral from a to b of sqrt(1+f'(x)^2)dx. After taking the derivative and simplifying, the integral becomes sqrt{1+[1/2x^(-1/2)*(1-x)]} dx. This can be further simplified by factoring to (1/2x-1/2 + 1/21/2)^2.
  • #1
gigi9
40
0
Calc help please!

Plz help me out w/ the problem below. Thanks.
Find the length of the specified arc of the given curve:
y=1/3sqrt(x)*(3-x), 0<=x <=3
Formular to find arc length is integral from a to b of sqrt(1+f'(x)^2)dx
I got to these steps below and not sure exactly wat to do next
***y'=1/6x^(-1/2)(3-3x)=1/2x^(-1/2)(1-x)
***V= integral from 0 to 3 of sqrt{1+[1/2x^(-1/2)*(1-x)])2} dx
=integral 0 to 3 of sqrt[1/2+1/4*x^-1+1/4*x] dx
 
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  • #2
V= integral from 0 to 3 of sqrt{1+[1/2x^(-1/2)*(1-x)])2} dx
=integral 0 to 3 of sqrt[1/2+1/4*x^-1+1/4*x] dx

1/(4x) + 1/2 + x/4 can be factorized to (1/2x-1/2 + 1/21/2)2
 
  • #3


To find the length of the specified arc, we can use the formula given in the problem: integral from a to b of sqrt(1+f'(x)^2)dx. In this case, a=0 and b=3, so we have:

L = integral from 0 to 3 of sqrt(1+(1/2x^(-1/2)(1-x))^2)dx

Next, we need to simplify the expression inside the square root:

1+(1/2x^(-1/2)(1-x))^2 = 1+1/4x^-1+1/4x

Substituting this into the original equation, we have:

L = integral from 0 to 3 of sqrt(1/2+1/4x^-1+1/4x) dx

Now, we can use the power rule for integration to solve this integral:

L = integral from 0 to 3 of (1/2+1/4x^-1+1/4x)^(1/2) dx

= (1/2+1/4x^-1+1/4x)^(3/2) / (3/2) from 0 to 3

= (1/2+1/4*3^-1+1/4*3)^(3/2) / (3/2) - (1/2+1/4*0^-1+1/4*0)^(3/2) / (3/2)

= (1/2+1/4+3/4)^(3/2) / (3/2) - (1/2+0+0)^(3/2) / (3/2)

= (2)^(3/2) / (3/2)

= 2^(3/2) / 3

Therefore, the length of the specified arc is 2^(3/2) / 3 units.
 

1. What is the formula for finding the length of an arc on a curve?

The formula for finding the length of an arc on a curve is L = rθ, where L is the length of the arc, r is the radius of the curve, and θ is the central angle of the arc in radians.

2. How do you find the central angle of an arc on a curve?

The central angle of an arc on a curve can be found by dividing the arc length by the radius: θ = L/r. This equation can also be rearranged to solve for the arc length or radius if the other two values are known.

3. What is the unit of measurement for the length of an arc on a curve?

The unit of measurement for the length of an arc on a curve is typically in units of length, such as meters or feet. However, if the radius is given in units of length, then the arc length will have the same units.

4. Can the length of an arc on a curve be negative?

No, the length of an arc on a curve cannot be negative. It is a physical distance and therefore cannot have a negative value.

5. Are there any other methods for finding the length of an arc on a curve?

Yes, there are other methods for finding the length of an arc on a curve, such as using calculus or numerical integration. These methods may be necessary for more complex curves or when the central angle is not known.

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