# Is there a way to multiply two integrals? [closed]

This would be a great breakthrough in our study at school.

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You need to give vastly more information about what you mean. Do you mean multiplying two definite integrals, like $$\left(\int_a^b f(x)\,dx\right)\left(\int_c^d g(x)\,dx\right)\quad?$$ If so, then you're just multiplying two numbers. –  Zev Chonoles Mar 31 at 7:13
The operations that are known are the ones you are being taught at school... many things would be very neat, but sadly don't work. –  vonbrand Mar 31 at 9:55

## closed as not a real question by Ittay Weiss, tomasz, Dennis Gulko, vonbrand, Dominic MichaelisMar 31 at 10:12

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I think you may be asking when does a product of the integrals equals the integral of the product. This is possible under certain conditions. Let's work with double integrals. And let the domain of integration $R$ be rectangular in whatever two-dimensional coordinate system we are integrating in. And the function which we are integrating must be separable in the two variables of integration. Then we have that
$$\begin{eqnarray} \int_R f(x,y)dA &=& \int_{[a,b]\times[c,d]}f(x,y)dydx\\ &=& \int_{[a,b]\times[c,d]}g(x)h(y)dydx\\ &=& \int_a^b \int_c^d g(x)h(y)dy dx\\ &=& \int_a^b g(x) \left(\int_c^d h(y) dy\right)dx\\ &=& \left(\int_a^b g(x)dx\right) \left(\int_c^d h(y) dy\right). \end{eqnarray}$$
But not with simple integrals: generally speaking, $\int f g \neq \int f \int g$ –  arbautjc Mar 31 at 9:20