Can Definite Integrals Be Divided Algebraically?

[drewfstr314]

Is there a general algebraic way to write the quotient of two definite integrals as one? I mean, what would be

\frac{\int_a^b f(s) ds}{\int_c^d g(t) dt}

Is it analogous to the product of integrals creating a double integral?

Thanks in advance!

 

[lugita15]

Let's see how a double integral of a product (of functions of different variables) can be written as a product of single integrals:\int^{d}_{c}\int^{b}_{a}f(s)g(t) dsdt = \int^{d}_{c}g(t)\left(\int^{b}_{a}f(s) ds\right) dt = \left(\int^{b}_{a}f(s) ds\right)\left(\int^{d}_{c}g(t) dt\right). You can verify that the same kind of thing doesn't work for quotients, because the integral of a reciprocal is not the reciprocal of the integral.