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The linear density of a rod of length $ 4 m $ is given by $ \rho (x) = 9 + 2 \sqrt{x} $ measured in kilograms per meter, where $ x $ is measured in meters from one end of the rod. Find the total mass of the rod.

$$46 \frac{2}{3} \mathrm{kg}$$

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Campbell University

Oregon State University

University of Nottingham

we have a question in which it is given that the linear density of Arad of length. The length of the rod is four metre and linear density. Little X. Is being given as nine plus 200 tax measuring kilograms per meter. So this is in kilograms per meter X in meters from one another rod. Okay, find the total mass of the rod. So if this is the Lord for up to X. This is a linear features, density. We know that linear channels then stage always must buy what we say must buy. Okay, X. Which is equal to nine plus 200 X. Okay, so D. M will be nine place 200 X in two D. X. Now we need to find the length of the door and the total mass of the length. So if we need we will be integrating it to get the to get that total mass. So if length is zero. So mass will be zero. If length is totally four m. So mass will be equal to total mass capital. And we should Right, okay, so this is mm equal to nine X plus two X. Raised to the power one by two plus one by one by two plus one. 0 to 4. Okay, nine X plus two X rays. To the power three by two, divide by three by two from 0 to 4. So this is nine x. place four by three X. Raised to the power three by two From 0-4. Now, if you plug in four Will be a parliament which is four. So 36 plus four x 3 and four raised to the power three x 2. So this is eight And zero will be zero everywhere. There is term containing acts 36. PL 32 by three. Okay. 108 miles 32 By three. That is 76 x three. So total mass of that Rod will be 76 x three. Katie, which will approximately be 26 5.33. Katie. Thank you.