Find arclength of curve; stuck trying to integrate radical

[Ryaners]

Homework Statement

Find the arclength of the curve x = ⅓(y²+2)^3/2 from y=0 to y=1.

Homework Equations

(Please forgive the crazy definite integral symbols - I'm taking a LaTex class tomorrow so hopefully I'll be able to communicate more clearly from then on..!)

arclength of curve = ∫_a^b √ (1 + f'(x)²) dx

The Attempt at a Solution

*Note: We haven't covered trig substitution in class yet & so I'm looking for a way to do this using u-substitution or another method.*

This is where I've gotten so far:

dx/dy = y√(y+2) = √(y³ + 2y²)

arclength = ∫₀¹ √(1 + [√(y³ + 2y²)]²) dy
= ∫₀¹ √(1 + y³ + 2y²) dy

Then I get stuck. I've tried the following substitution:
u = 1 + y³ + 2y²
du = 3y² + 2y

but that doesn't seem to help. I've also tried getting y in terms of x & integrating that but I reach a similar stumbling block with that approach too. Wolfram Alpha gives a nice clean answer of 4/3 which makes me think I'm missing something fairly obvious. Any pointers much appreciated!

 

[PeroK]

This is where I've gotten so far:

dx/dy = y√(y+2) = √(y³ + 2y²)

Any pointers much appreciated!

You may want to double check your differentiation.

 

[Ray Vickson]

Homework Statement

Find the arclength of the curve x = ⅓(y²+2)^3/2 from y=0 to y=1.

Homework Equations

(Please forgive the crazy definite integral symbols - I'm taking a LaTex class tomorrow so hopefully I'll be able to communicate more clearly from then on..!)

arclength of curve = ∫_a^b √ (1 + f'(x)²) dx

The Attempt at a Solution

*Note: We haven't covered trig substitution in class yet & so I'm looking for a way to do this using u-substitution or another method.*

This is where I've gotten so far:

dx/dy = y√(y+2) = √(y³ + 2y²)

arclength = ∫₀¹ √(1 + [√(y³ + 2y²)]²) dy
= ∫₀¹ √(1 + y³ + 2y²) dy

Then I get stuck. I've tried the following substitution:
u = 1 + y³ + 2y²
du = 3y² + 2y

but that doesn't seem to help. I've also tried getting y in terms of x & integrating that but I reach a similar stumbling block with that approach too. Wolfram Alpha gives a nice clean answer of 4/3 which makes me think I'm missing something fairly obvious. Any pointers much appreciated!

Your differentiation/arclength formula is incorrect; go back and check your algebra.

 

[Ryaners]

Aha! Thank you both for pointing out that error in computing the derivative - getting a nice factorizable expression now :)