Find the centroid of the area in the first quadrant bounded by the curve y=e^-x and the axes and the ordinate x=ln5

Answer 1

The area of the region, A=∫[0,ln5] e^-x dx = 4/5
So, using the normal formulas,

x̅ = 5/4 ∫ x f(x) dx = ∫[0,ln5] xe^-x dx
y̅ = 5/4 ∫ y f(y) dy = ∫[0,1] -y lny dy

You can do those using integration by parts.

Answer 2

Oops. confused y̅ with the moments from the y-axis.

y̅ = 5/4 ∫ 1/2 y^2 dx = ∫[0,ln5] 1/2 e^-2x dx