Need Help Integrating This Tricky Integrand

[royblaze]

Find the length of the curve:

r(t) = <2t, t², (1/3)t³>
r'(t) = <2, 2t, t²>

From bounds of t: 0 to 1.

So length = integral of the modulus of r'(t):

Integral of sqrt(t⁴+4t²+4)

I'm just dead stuck on how to attack it. I tried to make it integral of sqrt((t²+2)²), and then just getting rid of the square, but I'm feeling intrinsically unsure that that way will work.

Would setting a u = t⁴ help at all?

Any help is appreciated!

 

[micromass]

Find the length of the curve:

r(t) = <2t, t², (1/3)t³>
r'(t) = <2, 2t, t²>

From bounds of t: 0 to 1.

So length = integral of the modulus of r'(t):

Integral of sqrt(t⁴+4t²+4)

I'm just dead stuck on how to attack it. I tried to make it integral of sqrt((t²+2)²), and then just getting rid of the square, but I'm feeling intrinsically unsure that that way will work.

Yeah, sure, try it that way!

 

[royblaze]

I tried it and got a numerical value. So, I guess it's alright then.

Is it too much to ask for help on a different one? It's in the same form.

r(t) = <12t, 8t^3/2, 3t²>
r'(t) = <12, 12t^1/2, 9t>

Bounds of t = 0 to 1

Integral of sqrt(12² +(12t^1/2)² + (9t)²)dt

= integral sqrt(144 + 144t +81t²)dt.

I'm really just baffled; these are problems 1-10, which are technically supposed to be pretty simple. Am I just missing a glaringly easy way to integrate these things?

 

[micromass]

I tried it and got a numerical value. So, I guess it's alright then.

Is it too much to ask for help on a different one? It's in the same form.

r(t) = <12t, 8t^3/2, 3t²>
r'(t) = <12, 12t^1/2, 9t>

Bounds of t = 0 to 1

Integral of sqrt(12² +(12t^1/2)² + (9t)²)dt

= integral sqrt(144 + 144t +81t²)dt.

I'm really just baffled; these are problems 1-10, which are technically supposed to be pretty simple. Am I just missing a glaringly easy way to integrate these things?

Try to complete the square here. Prepare for a trigonometric substitution.

 

[royblaze]

I'm having trouble after completing the square. I don't remember too much about trigonometric substitution... :(

I factored out the 81, to move a constant out:

9 * (integral of sqrt(t² + (16/9)t +16/9))

So I took the square of half of 16/9:

9 * (integral of sqrt([t² + (16/9)t + (64/81)] + 80/81))

9 * (integral of sqrt((t+(8/9))² + 80/81)

EDIT: And just for questioning's sake; the bounds are from 0 to 1, so I'm guessing the answer itself shouldn't be hard to compute?

 

[micromass]

Factor out 80/81 and set

$$\tan^2{u}=\frac{81}{80}(t+\frac{8}{9})^2$$

 

[royblaze]

It might be the energy drink crash I'm getting here, but I don't see how the 80/81 can be factored out...

Sorry :/

 

[micromass]

$$\sqrt{x-a}=\sqrt{a}\sqrt{\frac{x}{a}-1}$$

Do something like this.

 

[royblaze]

So I get

a = 80/81

sqrt(a) * (sqrt((t + 8/9)²)/a + 1)

I can't see how the sqrt goes away. Was I supposed to change the sign in b/t the binomal squared term and the 1?

 

[micromass]

Now make an appropriate trigonometric substitution. Substitute tan in.

 

[royblaze]

Do you mean in the sense of the derivative of arctan? (1/(1+x²))?

If I substitute tan in,

u = tan(t+(8/9))²,

Is this the correct substitution? It seems different from yours.

 

[royblaze]

Ah ha! I found it. By factoring out scalars from the vectors, I made my life a ton easier.

I found it out after consulting my textbook.

Thanks though :)