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

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

#### Topics

Applications of Integration

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### Video Transcript

problem worse to shoulder integral A to B C X plus tee times. If affects, the X is equal to see expert policy internal i to be f off x t x. Now we know that area often region on record is integral f effects. B X referred to know that it's part of the region is one or area times x f off x e x Combining these we see that then of export times area is equal to inter go x ffx the X And since we know what areas, we can see that then export times in Tegel f x The X is equal to integral x ffx d Thanks. All right now let's multiply both sides off this equation by sea and we get See Time's export in jungle ffx T X is equal to, since he's a cost Angeles Right inside the integral you have C times Thanks times FX, the eggs and balance and in Tegel D F. Effects. The Extra Boat Science of e half See export Internal ffx DX plus de inter go F effects. D X is equal to inter. Go see ETS f effects the X plus in Tegel de ffx t X. Now we can group this one as sense in Trigo FX. The exes are coming on the left hand side. We can write this one as c ex parte plus deed times integral effects D x is it going to now? He had two intervals and we're into American would suspect D X so we can let this one inside. They want integral, and we can write that won us see, uh, X plus de times if effects he x and up. Let's compare this with what is given here. Then we can see that we just prudent Oh.

#### Topics

Applications of Integration

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