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Since $\int_ {D} f_- dA$ is nonnegtive  $$\int_ {D} f_+ dA - \int_ {D} f_- dA \leq \int_ {D} f_+ dA + \int_ {D} f_- dA$$  Since $f_+ ≥ 0$ and $f_− ≥ 0$ are integrable over $D$  $$\int_ {D(n)} f_+ dA + \int_ {D(n)} f_- dA =\int_ {D(n)} (f_+ + f_-) dA$$  By the properties of limits of increasing sequence D(n), we know $\int_ {D(n)} (f_+ + f_-) dA$ converges  so $$\int_ {D} f_+ dA + \int_ {D} f_- dA =\int_ {D} (f_+ + f_-) dA$$  By the equation $f(x,y) = f_+(x,y)− f_−(x,y)$, we got $$\int_ {D} f dA \leq \int_ {D} \left|f \right|dA$$  (b) In the same way, we apply (a) to the functions − f to get  $$\int_ {D} -f dA \leq \int_ {D} \left|-f \right|dA= \int_ {D} \left|f \right|dA$$  (c)By the properties of limits and the equation $\int_ {D(n)} -f dA =$\int_ {D(n)} f dA $,  we get$$- \int_ {D} f dA \leq \int_ {D} \left|f \right|dA$$  (d)If $b\leq a$ and$ $−b\leq a$ then$|b|\leq a.$  From (a), we got $$\int_ {D} f dA \leq \int_ {D} \left|f \right|dA$$,  From (b) and (c), we got $$- \int_ {D} f dA \leq \int_ {D} \left|f \right|dA$$  Therefore, we can conclude that  $$\left| \int_ {D} f dA\right| \leq \int_ {D} \left|f \right|dA$$