Solve Integral Problem: y=t^2, x=t^3

[jdc15]

Homework Statement

This isn't really out of a textbook but rather is my own inquiry, but here goes:

y=t²
x=t³

Find $$\int{y}dx$$

Homework Equations

Integration rules maybe?

The Attempt at a Solution

I'm pretty sure the answer is $$y\int{}dx=y(x+C)=t^2(t^3+C)$$ but why can't I go $$dx=3t^2dt$$ and then $$\int{y}dx=3\int{t^4}dt=3t^5/5+C$$? Obviously one is right and one is wrong (or both wrong) but why is this so?

Thanks in advance.

 

[tiny-tim]

hi jdc15!

(have an integral: ∫ )

no, the first is completely wrong: you've treated y as a constant: if you're going to do it without t, you need to express y as a function of x, which is … ?

(and the second is right)

 

[tjackson3]

tiny-tim is right. Here's a more conceptual look at it: Consider what you're saying when you're talking about $\int\ y\ dx$. You're talking about different values of y as x changes. Now, without the parametrization at the beginning, what you did would have been correct; however, in order for x to change, t must also change.

 

[jdc15]

Alright thank you very much. This question relates to a physics problem I have which I might post later if I can't figure it out. Unfortunately this conflicts with what a TA told me. Oh well I'll figure it out. Thanks again.

 

[micromass]

What did the TA tell you. Maybe he interpreted it another way?

 

[jdc15]

Well, when I say "conflict" I really mean seems to conflict haha. Here was the original problem:

a) Consider a driven mass-spring system with viscous friction using the notation of the lecture of Oct. 29, available on Vista. [The driving frequency is ω, the natural frequency is ω0, the friction force is -cv, the mass is m, the spring constant is k, the driving force is kD sin (ω t). Note that the phase Φ is negative, between 0 and π .] Write a formula for the rate of energy loss due to the friction force, once the steady state has been reached, as a function of time, t.

b) Write a formula for the rate at which the driving force is doing work on the mass-spring system, once the steady state has been reached, as a function of time, t.

c) Find the total energy loss due to friction over one period of the oscillations and also the total work done by the driving force over one period. Check whether or not they are equal. To do this problem you need to use the fact that the integral of a sine or cosine function over one period is zero. This integral is simply the average value over one period divided by the period. This is clearly zero if you think about what a cosine or sine function looks like.


Bascially we have an oscillating mass on a spring, with viscous friction, and a driving force kDsin(wt). (w is omega but typing w is faster). We have the formula $$W=\int{F}\cdot dr$$. In this case it simplifies to $$\int{F}dr$$. Both F and r depend on time, with F being given by F=-cv=-cdr/dt and r being Asin(wt+phi), with A having been predetermined in our lecture to be some constant function of m, k, c and w. What she told me to do was to go $$\int{F}dr=F\int{}dr=Fr$$ but that doesn't seem right to me.

Edit: I'm going to move this to the physics section
Edit 2: Here is a link to the new post https://www.physicsforums.com/showthread.php?p=2967771#post2967771"