I have not realized these problems are really tough, due to the nature of the bounded area.

For the centroid, . xbar = [2*(2.58492) + 2.739105283(7.029682312)] / [2*(2.92847) + 7.029682312 ] = 1.8953671

for ybar it is at the axis of symmetry which is the x axis, so ybar is zero.

so centroid is (xbar, ybar) = (1.8953671, 0)

(interestingly, the centroid is outside the area, and this is common for unusual shapes like this (crescent) horshoes, etc.) . (By the way, to make it simpler to compute for the centroid, I divided the crescent using the line x = 2, so the bounded region was divided by three subparts) .

I am sorry I will need more time to answer for the remaining problems, as this will require more analysis.

there are really tough and take time to solve. I technically need them in 2 hours and half but I also need some time to write them down before the submission. Please try to finish them asap

Hi, I think I need more time, (maybe at least a day) to answer the remaining question. (right now it is early dawn in our place). . For the first part (centroid) . solution is (when partitioned by x = 2) . area of subpart 1 (the upper and the lower part has the same area) . integral (sqrt(8 -(x-1)^2) - sqrt (4 - x^2) dx from -1.5 to 2 = 2.92847 (area) . integral x(sqrt(8 -(x-1)^2) - sqrt (4 - x^2) dx from -1.5 to 2 = 2.58492 .

For the middle part: . Area = integral 2*sqrt (8 - (x-1)^2)dx from 2 to (1+sqrt8) = 7.029682312 . and

. integral 2x*sqrt (8 - (x-1)^2)dx from 2 to (1+sqrt8) = 19.3765217636757 . . So . Centroid of the Crescent region: . xbar = [2*(2.58492) + 19.3765217636757] / [2*(2.92847) + 7.029682312 ] . = 1.90479407 . for ybar it is at the axis of symmetry which is the x axis, so ybar is zero. . so centroid of the crescent shape is (xbar, ybar) = (1.90479407, 0)

will try again, part b and c I think is possible, I am still figuring out for the last part ( with reference to z) will update you later (still early dawn here)

It's really stressing me out that my solutions are never submitted on time. If you can't do it in 12 hours pleases let me know now. Other than that I will probably request a refund and I don't want to waste your time or mine.