Running the Limits Backwards

Two sensors record signals $f$ and $g$ over the first $7$ seconds of a test run. The raw traces were discarded, and only these four signed areas were kept in the log:

Integral	Logged value
$\int_{0}^{3} f(x)\,dx$	$8$
$\int_{0}^{7} f(x)\,dx$	$3$
$\int_{0}^{3} g(x)\,dx$	$10$
$\int_{3}^{7} g(x)\,dx$	$-4$

You may use the fact that a definite integral passes straight through sums and constant multiples:

$\int_{a}^{b}\left[\alpha f(x) + \beta g(x)\right] dx = \alpha \int_{a}^{b} f(x)\,dx + \beta \int_{a}^{b} g(x)\,dx$

Using only the logged values and the properties of the definite integral, evaluate

$I = \int_{7}^{0} \left[\,2f(x) - 3g(x)\,\right] dx$

What is the value of $I$ ? Your answer is a whole number.

Running the Limits Backwards

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