Question 4 010 pts 5 99 compute the center of mass barx on a...

Question 4 0/10 pts 5 99 Compute the center of mass $\bar{x}$ on a rod of density $\rho(x) = 4x^3 + xe^{-x}$ with $x \in (0,3)$. $\bar{x} = \boxed{\phantom{1234567890}}$ Round to 4 decimal places. Use technology as needed. Question Help: Video Message instructor Submit Question

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00:01 Hi there, so for this problem to find the centroid of the region bounded by the following curves, that will be y equals to 4 times x square, plus 2 times adds, and that adds equals to, well, y, sorry, y equals to 0, and the vertical line adds equals to 8.

00:30 We need to compute the coordinates at y bar and y bar using the formulas of the centroid of a region so these formulas are the following one divided by a times the integral from a to b of x times the function f of x that we are given and this integrated over x and y bar is just one divided by a the integral from a to b of the function f of x to the squared and this divided by two, this integrated over x.

01:05 So what we need to do first is to determine a.

01:08 A is the area of this region.

01:10 So to determine the area of this region, that will be the integral from 0 to 8 because we start at 0 and n at x equals to 8.

01:29 So this will be the integral of four times x squared.

01:32 Plus 2xxxxx integral over x the solution of this integral is square is that will be 4 divided by 3 times x to the 3 plus x squared now we evaluate this from 0 to 8 now we just do the evaluation in here we don't we just need to evaluate this at 8 because at 0 we will obtain just simply 0 the value that we obtained from this is about 2 ,200 140 divided by 3.

02:05 Now let's do add x bar.

02:08 So that will be 1 divided by 8.

02:15 This is the integral from 0 to 8 of the function that we're given times x.

02:26 So that will give us 4 times x to the 3 plus 2 times x squared integrated over x.

02:31 So we just solve this integral...