Integrating parametric equations

1. May 12, 2015

bigplanet401

1. The problem statement, all variables and given/known data

Why does
$\int_a^b \, y \; dx$
become
$\int_\alpha^\beta \, g(t) f^\prime(t) \; dt$
if x = f(t) and y = g(t) and alpha <= t <= beta?
2. Relevant equations

Substitution rule?

3. The attempt at a solution

I'm not sure how y = y(x) in the integrand turns into g(t). Isn't y a function of x in the first expression? How do they go from y(x) to g(t)?

2. May 12, 2015

Staff: Mentor

Substitution.
Replace y by g(t). What should you replace dx by?

3. May 12, 2015

bigplanet401

If x = f(t), dx = f'(t) dt. I understand that part.

But in
$$\int y \; dx$$

isn't y = y(x) a function of x? We'd then have y = y(x) = y[x(t)]. How can we just let y = g(t) and get the resulting expression in t?

4. May 12, 2015

Staff: Mentor

No, not according to the problem description you wrote, which says y = g(t). x is a different function of t.
Because it is given that y = g(t).

5. May 12, 2015

Zondrina

You want to show:

$$\int_a^b y \space dx = \int_\alpha^\beta g(t) f^\prime(t) \space dt$$

When given:

$$x = f(t)$$
$$y = g(t)$$

Write $\frac{dx}{dt} = f^\prime(t)$; You also know $y$.

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