Evaluate a definite integral with two variables?

[LilTaru]

Homework Statement

Evaluate.

$$\int$$ y²(1 - y³/a²)^-2 dy

Upper limit = 0
Lower limit = -a

Homework Equations

The Attempt at a Solution

The a's make me really confused and I have no idea how to even start this question.
What I started with was:

Set u = 1 - y³/a²
du = (2y² - 3y²a)/a³

Then I know you are supposed to make the integral contain only u's, but I am so confused! Am I on the right track? If so, where do I go from here? If not, can someone lead me on the right track?

 

[vela]

Just treat a as a constant.

 

[Char. Limit]

It might also make it easier to multiply top and bottom by a⁴.

 

[LilTaru]

I think that is the part that confuses me... If a is constant, how do we take the antiderivative?

 

[Char. Limit]

With respect to y. Just treat a as you would treat a 3, or a pi.

 

[vela]

Just for clarification, is the problem

$$\int_{-a}^0 y^2\left(\frac{1-y^3}{a^2}\right)^{-2}\,dy$$

or

$$\int_{-a}^0 y^2\left(1-\frac{y^3}{a^2}\right)^{-2}\,dy$$

? I'm guessing from what you wrote down for du, it is the former, but what you wrote in your original post, with the lack of parentheses, means the latter.

 

[LilTaru]

It's the latter option. I always forget to include parentheses!

 

[vela]

Oy, brain fart on my part. I think your du is consistent with the latter integral if you're making the mistake I think you're making. How you wrote the integral in your original post is fine. It's just that a lot of posters leave out parentheses so when I see problems written like this, it's a question that immediately pops into my mind.

To be a bit more explicit about what we're telling you...

Consider the function f(x)=9x². When you differentiate it, you don't look at it as a product of 9 and x² and apply the product rule, right? You just say the 9 is a constant so you can just pull it out front and then just differentiate the x². Similarly, if you had g(x)=x²/3², you wouldn't use the quotient rule. You'd just pull the constant 1/3² out front and differentiate the x². When we say treat a as a constant, we mean you can just pull it out front. You don't treat is as a function of x (or y in your case) even though it's written using a letter rather than numbers.

 

[LilTaru]

Ohhhhh! Wow... now it makes so much more sense... So, I can just pull 1/a² out of the integral and than it becomes:

$$\int$$ y²(a² - y³)^-2 ?

Since we need to factor out 1/a² from both 1 and y³/a².
And then continue with substitution? This looks like it will work to me...

 

[Char. Limit]

Ohhhhh! Wow... now it makes so much more sense... So, I can just pull 1/a² out of the integral and than it becomes:

$$\int$$ y²(a² - y³)^-2 ?

Since we need to factor out 1/a² from both 1 and y³/a².
And then continue with substitution? This looks like it will work to me...

Watch out. Because the 1/a^2 is inside another ², you need to pull out 1/a^4.

 

[vela]

Well, you can't just erase it altogether, and you must still follow the rules of algebra. After you pull the factor out front, you should have

$$a^4\int_{-a}^0 y^2(a^2-y^3)^{-2}\,dy$$

 

[HallsofIvy]

Yes, use the obvious substitution.

(You forgot the "dy" which is very important- for one thing, it was the "dy" that told you that, at least for the purposes of this integration, a is a constant.)

 

[LilTaru]

Yes. Sorry. I meant with the dy and with 1/a² at the front. And thanks for catching it was 1/a⁴. Forgot about the squaring! Thanks a lot everyone! I think I can solve it now!

 

[LilTaru]

Solved it! I think it's correct... Got the answer as
(a² - a⁶ + a⁷)/3

Since:

Set u = a^2 - y^3; du = -3y^2 dy
Let G(u) = 1/3(u)^-1 (the antiderivative)
a^4 * G(0) - G(-a) (evaluation of the integral)
= a^4 * (1/3a^2) - ((a^2 + a^3)/3)
= (a² - a⁶ + a⁷)/3

Is this correct?