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Multivariable Calculus

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Difficult

Separated Integral

MVCALC-ULGEAY

Let the double variable function $C(x,y)=f(x)g(y)$, both $f$ and $g$ are continuous functions for all real numbers $x$ and $y$ respectively. Assume $a,b,c,d$ are constants, which of the following equations are true:

$$\begin{align*} &I:\quad \int_a^b \int_c^d C(x,y)dy dx=\big(\int_a^b f(x)dx\big) \big(\int_c^d g(y)dy \big) \\\ &II:\quad \big(\int_a^b f(x)dx\big)^2=\int_a^b \int_a^b f(x)f(y)dydx \\\ &III: \quad \int_a^b f(x)\big(\int_c^d g(y)dy\big) dx = \int_c^d g(y)\big(\int_a^b f(x)dx\big) dy \\\ \end{align*}$$

A

Only $I$

B

Only $III$

C

Only $I, III$

D

$I, II, III$

E

None of the above