Integrating parametric equations

[bigplanet401]

Homework Statement

Why does
<br /> \int_a^b \, y \; dx<br />
become
<br /> \int_\alpha^\beta \, g(t) f^\prime(t) \; dt<br />
if x = f(t) and y = g(t) and alpha <= t <= beta?

Homework Equations

Substitution rule?

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)?

 

[Mark44]

Homework Statement

Why does
<br /> \int_a^b \, y \; dx<br />
become
<br /> \int_\alpha^\beta \, g(t) f^\prime(t) \; dt<br />
if x = f(t) and y = g(t) and alpha <= t <= beta?

Homework Equations

Substitution rule?

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

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)?

 

[bigplanet401]

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

But in
<br /> \int y \; dx<br />

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?

 

[Mark44]

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

But in
<br /> \int y \; dx<br />

isn't y = y(x) a function of x?

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

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?

Because it is given that y = g(t).

 

[STEMucator]

Homework Statement

Why does
<br /> \int_a^b \, y \; dx<br />
become
<br /> \int_\alpha^\beta \, g(t) f^\prime(t) \; dt<br />
if x = f(t) and y = g(t) and alpha <= t <= beta?

Homework Equations

Substitution rule?

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)?

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 $y$.