Centroid Position of a Lamina Bounded by a Curve

[Alex_Neof]

Homework Statement

A lamina is bounded by the x-axis, the y-axis, and the curve $y = 4 -x^2.$ Determine the centroid position $(\bar{x},\bar{y})$ of the lamina.

Homework Equations

$A = \int_a^b (f(x) - g(x)) dx$ (Area)

$\bar{x} = \frac{1}{A}\int_a^b x(f(x) - g(x)) dx$

$\bar{y} = \frac{1}{A}\int_a^b \frac{1}{2}(f(x)^2 - g(x)^2) dx$

The Attempt at a Solution

I made a sketch and determined $a = 0$ and $b = 2$ for the limits.

Then just plugged into the above equations.

With this I determined the area to be $A=16/3$

$\bar{x} = \frac{3}{4}$

$\bar{y} = \frac{8}{5}$

Therefore centroid position is $(\frac{3}{4},\frac{8}{5})$

Could someone kindly verify this?

 

[STEMucator]

Everything looks good at a glance.

 

[Alex_Neof]

It is correct, thank you . I used an online calculator to verify it. I'll write out a solution for any future viewers.