# Integration Factor Constant Rule for Double Integral

I know that for the situation where there is a constant c on a function I can factor it from the integrand. So,

$\int c f(x) dx$
$c \int f(x) dx$

What about a double integral? Would it be like

$\iint c f(x,y) dx dy$
$c \iint f(x,y) dx dy$

Is that correct?

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In a word: yes. – J. M. Oct 28 '12 at 5:00
To see that this is so, imagine that $f$ is constant so that the double integral is the volume of a rectangular prism. If the situation is clear there, next, think about the definition of the integral as a limit of Riemann sums (if you know this, learn it if you don't) and general case should become clear. – Jonah Sinick Oct 28 '12 at 5:59

Yes, by definition $$\iint f(x,y)dxdy = \lim_{(\Delta x_i,\Delta y_j) \to (0,0)} \sum\sum f(\xi_i,\psi_j)\Delta x_i \Delta y_j$$

Now $$\iint cf(x,y)dxdy = \lim_{(\Delta x_i,\Delta y_j) \to (0,0)} \sum\sum cf(\xi_i,\psi_j)\Delta x_i \Delta y_j$$

Now, use the properties of Sums and limits to obtain:

$$\lim_{(\Delta x_i,\Delta y_j) \to (0,0)} \sum\sum cf(\xi_i,\psi_j)\Delta x_i \Delta y_j = c\left(\lim_{(\Delta x_i,\Delta y_j) \to (0,0)} \sum\sum f(\xi_i,\psi_j)\Delta x_i \Delta y_j\right)$$

Therefore,

$$\iint cf(x,y)dxdy = c\iint f(x,y)dxdy$$

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