1. centroid

Find the centroid of the region bounded by y = lnx, y = 0, 1 >= x >= e

What I did was found the area by taking the integral of lnx and evaluating it @ 1 and e.
Then I put it in the formula to find x bar and y bar.
I got an undefined numbers for x and -1/3 for y.

Does anyone know how to correctly solve this problem?

2. Originally Posted by sleepiiee
Find the centroid of the region bounded by y = lnx, y = 0, 1 >= x >= e

What I did was found the area by taking the integral of lnx and evaluating it @ 1 and e.
Then I put it in the formula to find x bar and y bar.
I got an undefined numbers for x and -1/3 for y.

Does anyone know how to correctly solve this problem?
$\bar x =\frac{\int_{1}^{e}x\ln(x)dx}{\int_{1}^{e}\ln(x)dx }$

Both of the above are integrated by parts to give

$\bar x =\frac{\frac{1}{4}+\frac{1}{4}e^{2}}{1}$

$\bar y =\frac{\int_{1}^{e}[\ln(x)]^2dx}{\int_{1}^{e}\ln(x)dx}$

The top in integration by parts twice and the bottom is 1 from above to give

$\bar y =\frac{e-2}{1}=e-2$