Integrating parametric equations

In summary: When given:$$x = f(t)$$$$y = g(t)$$$$\int_\alpha^\beta y \space dx = \int_\alpha^\beta g(t) f^\prime(t) \space dt$$
  • #1
bigplanet401
104
0

Homework Statement



Why does
[itex]
\int_a^b \, y \; dx
[/itex]
become
[itex]
\int_\alpha^\beta \, g(t) f^\prime(t) \; dt
[/itex]
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)?
 
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  • #2
bigplanet401 said:

Homework Statement



Why does
[itex]
\int_a^b \, y \; dx
[/itex]
become
[itex]
\int_\alpha^\beta \, g(t) f^\prime(t) \; dt
[/itex]
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?
bigplanet401 said:

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)?
 
  • #3
If x = f(t), dx = f'(t) dt. I understand that part.

But in
[tex]
\int y \; dx
[/tex]

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
bigplanet401 said:
If x = f(t), dx = f'(t) dt. I understand that part.

But in
[tex]
\int y \; dx
[/tex]

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.
bigplanet401 said:
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).
 
  • #5
bigplanet401 said:

Homework Statement



Why does
[itex]
\int_a^b \, y \; dx
[/itex]
become
[itex]
\int_\alpha^\beta \, g(t) f^\prime(t) \; dt
[/itex]
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##.
 

Related to Integrating parametric equations

1. What are parametric equations?

Parametric equations are a set of equations that describe the relationship between two variables in terms of a third variable, typically represented by the parameter t. They are commonly used in mathematics and physics to represent curves and surfaces in a coordinate system.

2. How do you integrate parametric equations?

To integrate parametric equations, you first need to express the functions in terms of the parameter t. Then, you can use the substitution method to replace the parameter with a variable, and apply standard integration techniques to evaluate the integral. The result will be a function of the variable, rather than the parameter t.

3. What is the purpose of integrating parametric equations?

Integrating parametric equations allows us to find the area under a curve or the volume of a solid of revolution in a parametric coordinate system. It also helps in solving various problems in physics and engineering, such as calculating work and fluid flow.

4. What are the limitations of integrating parametric equations?

One limitation of integrating parametric equations is that it can be more complex and time-consuming than integrating standard functions. Additionally, some parametric equations may not have closed-form solutions, making it challenging to find an exact answer through integration.

5. Are there any applications of integrating parametric equations in real life?

Yes, integrating parametric equations has many practical applications in real life. For example, it is used in designing roller coasters, analyzing projectile motion, and predicting the behavior of moving objects in physics. It is also used in engineering to model the movement of robots and vehicles.

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