# Using the Product of the Integrals is integrals of Product Magic in multivar

• Quantumpencil
In summary, the author is questioning why it is always valid to do the integrals Griffiths does in his electrodynamics textbook where he pulls out dphi and multiplies it by that integral.

#### Quantumpencil

Using the "Product of the Integrals is integrals of Product" Magic in multivar

This isn't actually a question on a homework problem, but It is coursework related so I thought it belonged here. I've noticed a lot of magic in Griffiths and other textbooks, where say, in integrating in spherical co-ordinates, you simply do the theta, phi, and r integrals, and them multiply them all together...

but I thought this was not a valid way to do integration... like it makes sense to me, but I can't justify to myself why it would always be true that one could do this... is it maybe only true when the function your integrating doesn't depend on all three variables, or you can get it into a form where the product of these things don't?

For instance, Int (e^-x^2-y^2)dxdy... across all of R^2 this integral of this is equal to the integral of e^-x^2 dx times from 0-> infinity the integral of e^-y^2 dy multiplied together from 0 to infinity... Why? and why is it legit to do this "pulling out dphi and multiplying by that integral" like Griffiths does in his electrodynamics?

$$\int \int e^{-x^2-y^2}dxdy=\int \int e^{-x^2}e^{-y^2}dxdy=\int e^{-y^2} \left(\int e^{-x^2}dx \right)dy=\int e^{-y^2} dy \int e^{-x^2} dx$$

Step 2 to 3: $e^{-y^2}$ is a constant with respect to x, so you can take it in front of the integral.
step 3 to 4: $\int e^{-x^2}dx$ is a constant with respect to dy, so you can take it in front of the integral.

In Griffiths you're probably talking about integrals like:

$$W=\frac{\epsilon_0}{2}\int_{all space}E^2 dV=\int_0^R \int_0^\pi \int_0^{2\pi} E^2 r^2 \sin \theta d\phi d\theta dr= \int_0^R E^2 r^2 dr \int_0^\pi \sin \theta d\theta \int_0^{2\pi} d\phi$$

This is because, E usually has a dependence on r alone, had it been $\vec{E(r,\theta,\phi)}$ this would only have worked if $\vec{E(r,\theta,\phi)}=E(r)E(\theta)E(\phi)$

For example:
$$\int \int 2 \cos (xy) \sin (xy)\, dxdy \neq 2 \int \cos (xy)\, dx \int \sin (xy)\, dy$$

Try it out.

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If the limits of integration are constant and the function is "separable", f(x,y)= g(x)h(y), that is valid.

$$\int_{x= a}^{b}\int_{y= c}^d g(x)h(y) dydx= \int_{x=a}^f g(x)\left(\int_{y= c}^d h(y)dy\right) dx$$
Since h is a function of y only and the limits of integration, c and d, are constant, the integration inside the parentheses is a constant and so can be taken out of the second integral:
$$\left(\int_{y= c}^d h(y)dy\right)\left(\int_{x= a}^b g(x)dx\right)$$

Great; Thanks guys!

## 1. What is the Product of the Integrals?

The Product of the Integrals is a mathematical concept used in multivariable calculus. It is the result of multiplying two integrals together, where each integral represents the area under a curve in a given region.

## 2. How is the Product of the Integrals used?

The Product of the Integrals is often used in multivariable calculus to find the volume of a three-dimensional region by breaking it down into two-dimensional slices and integrating the area of each slice. It can also be used to find the area of a surface by integrating the length of each curve on the surface.

## 3. What is the difference between using the Product of the Integrals and the Product Rule?

The Product of the Integrals is a concept used in calculus, while the Product Rule is a formula used in differential calculus to find the derivative of a product of two functions. They are two different techniques used for different purposes.

## 4. Can the Product of the Integrals be applied to more than two integrals?

Yes, the Product of the Integrals can be extended to any number of integrals, not just two. This is known as the Multiple Product Rule and is often used in higher-level calculus courses.

## 5. Are there any limitations to using the Product of the Integrals?

Like any mathematical concept, the Product of the Integrals has its limitations. It can only be used when the integrals are independent of each other and when the region of integration is a simple shape, such as a rectangle or a circle. It may also be more challenging to use the Product of the Integrals when dealing with more complex regions or functions.