Question about triple integrals

[cristi92b]

Can I write the following product of integrals as a triple integral? Is there a general formula for doing this?

\(\displaystyle \int_{0}^{1}dx\int_{-\sqrt{1-x^{2}}}^{\sqrt{1-x^{2}}}dy\int_{0}^{a}dz\)

 

[daon2]

Can I write the following product of integrals as a triple integral? Is there a general formula for doing this?

\(\displaystyle \int_{0}^{1}dx\int_{-\sqrt{1-x^{2}}}^{\sqrt{1-x^{2}}}dy\int_{0}^{a}dz\)

Yes but you have to be careful, because as written, the product of the integrals is actually a function of x.

\(\displaystyle \displaystyle \int_0^a \int_{-\sqrt{1-x^2}}^{\sqrt{1-x^2}} \int_0^1 dxdydz\)

You may interchange the y-integral and z-integral as well (one does not depend on the other). But the x-integral should always be evaluated before the y-integral, otherwise you will get a numerical answer instead.

 

[cristi92b]

Thank you!