Double Integration change of variables

• Economist08
In summary, the conversation discusses a transformation involving displacement and optimal paths using calculus of variations. The only difference between the two integrands is the swapping of variables. However, the range of integration must be carefully chosen in order to correctly swap the variables. The conversation ends with a welcoming message to the new member.
Economist08
Hi
I just cannot understand the following transformation, where $$\phi(t)$$ is the displacement of an optimal path using standard calculus of variations. All functions are defined between 0 and T. $$\phi$$ equals zero at 0 and T. r is some discount rate, e it the Euler number, t is time.

$$\int^{T}_{0}\theta(y(t))e^{-rt}\int^{t}_{0}\phi(\tau)d\tau]dt$$

This should be equal to

$$\int^{T}_{0}\int^{T}_{t}\theta(y(\tau))e^{-r\tau}d\tau]\phi(t)]dt$$

If anybody knows the answer, I would be very happy to get some help here.

Economist08 said:
Hi
I just cannot understand the following transformation, where $$\phi(t)$$ is the displacement of an optimal path using standard calculus of variations. All functions are defined between 0 and T. $$\phi$$ equals zero at 0 and T. r is some discount rate, e it the Euler number, t is time.

$$\int^{T}_{0}\theta(y(t))e^{-rt}\int^{t}_{0}\phi(\tau)d\tau]dt$$

This should be equal to

$$\int^{T}_{0}\int^{T}_{t}\theta(y(\tau))e^{-r\tau}d\tau]\phi(t)]dt$$

If anybody knows the answer, I would be very happy to get some help here.

I think you're being blinded by the detail.

Look carefully and you'll see that the only difference in the two integrands is that t and $$\tau$$ have been swapped over.

Now, you can always change the names of the variables. For example, ∫f(x)dx = ∫f(y)dy, or ∫∫f(x,y)dxdy = ∫∫f(z,w)dzdw.

In your example, x = w and y = z: ∫∫f(x,y)dxdy = ∫∫f(y,x)dydx.

The only place you have to be careful is in choosing the range of integration.

It is the triangular area 0 < $$\tau$$ < t; 0 < t < T.

In other words, all pairs of t and $$\tau$$ with $$\tau$$ < t < T.

Swapping t and $$\tau$$ gives: all pairs of t and $$\tau$$ with t < $$\tau$$ < T, or:
t < $$\tau$$ < T; 0 < t < T.

Welcome to PF!

Economist08 said:

You're very welcome!
Which reminds me, I forgot to say …

Welcome to PF!

1. What is the concept of double integration?

Double integration is a mathematical process used to find the area under a surface or volume between two curves. It involves integrating a function twice, first with respect to one variable and then with respect to the other variable.

2. What is change of variables in double integration?

Change of variables in double integration is a technique used to simplify the integration process by substituting the original variables with new ones. This is done to make the limits of integration easier to work with or to transform the integrand into a simpler form.

3. How do you perform a change of variables in double integration?

To perform a change of variables in double integration, you first need to identify the appropriate substitution and then apply the appropriate transformation formula for the new variables. This will change the limits of integration and the integrand, making the integration process easier.

4. Why is change of variables useful in double integration?

Change of variables is useful in double integration because it can simplify the integration process and make it easier to solve. It can also help in evaluating integrals that are difficult or impossible to solve using traditional methods.

5. What are some common examples of double integration with change of variables?

Some common examples of double integration with change of variables include finding the area between two curves, calculating the volume under a surface, and finding the centroid of a region. It is also commonly used in topics such as physics, engineering, and economics to solve various problems.

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