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If $ \bar{x} $ is the x-coordinate of the centroid of the region that lies under the graph of a continuous function $ f $, where $ a \le x \le b $, show that

$$ \int_a^b (cx + d) f(x) dx = (c \bar{x} + d) \int_a^b f(x) dx $$

$(c \overline{x}+d) \int_{a}^{b} f(x) d x=\int_{a}^{b}(c x+d) f(x) d x$

Applications of Integration

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