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

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

Calculus 2 / BC

Chapter 8

Further Applications of Integration

Section 3

Applications to Physics and Engineering

Applications of Integration

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If $\overline{x}$ is the $…

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Area and the centroid Let …

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If $f$ is continuous on $\…

This problem were asked to show it integral a to b c x, plus t times f of x. Dx is equal to c x, part plus t integral i to be f of x d x. Now we know that area of a region under cure is integral f of x d x, be further known at x. Part of a region is 1 over area times x, f of x d x. Combining these, we see that an x power times area is equal to integral x, f of x dx, and since we know what area is, we can see that the x bar times, integral f of x d x, is equal to integral x, f of x d X, all right now, let's multiply both sides of this equation by c, and we get c times x, bar integral f of x d x, is equal to since c is a constant, less writer inside the integral we have c times x, times, f of x, dx And not at integral d f of x d x to both sides. So we have c x, bar integral f of x, dx plus d, integral f of x. Dx is equal to integral c x, f of x d, x, plus integral d f of x d x. Now we can group this 1, as since, integral f of x d x are common. On the left hand side, we can write this 1 as c x. Bar plus d times, integral f of x, dx is equal to now we have 2 integral and we're integrating with respect to dx. So we can write this 1 inside 1 integral and we can write that 1 as c x, plus d times f of x, dx and now, let's compare this with what is given here and we can see that we just product.

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