find coordinates of the centroid: y = e^x, y=0 , x=0, x=3

Answer 1

First we get area A = ∫₀³e^xdx = e^x₀³ = e³ - e⁰ = e³ - 1;

Momentum S_y = ∫∫_Axdxdy = ∫₀³y₀^{e^x}xdx = ∫₀³xe^xdx = (xe^x - e^x)₀³ = (3e³ - e³) - (0 - e⁰) = 2e³ + 1

Momentum S_x = ∫∫_Aydxdy = ∫₀³y²/2₀^{e^x}dx = ∫₀³1/2(e^2x - 1)dx = (1/4e^2x - 1/2x)₀³ = (1/4e⁶ - 3/2) - (1/4) =

e⁶/4 - 7/4.

Now coordinates of centroid: (x_C, y_C)

x_C = S_y/A= (2e³ + 1)/(e³ - 1) = 2.1572

y_C = S_x/A = (e⁶/4 - 7/4)?(e³ - 1) = 5.1928