Sketch the region bounded by the curves, and visually estimate the location of the centroid. y = e^x, y = 0, x = 0, x = 4

Question

Trying to find xbar and ybar coordinates

xbar>0.5 y bar <1 is this correct?

A=∫(0 to 1)exdx

i can ntegrate im just trying to set this up correctly.

Any tips my friends?

Answer 1

My = ∫ [a to b] x [f(x) - g(x)] dx

Mx = 1/2 ∫ [a to b] [f(x)^2 - g(x)^2] dx

M = ∫ [a to b] [f(x) - g(x)] dx

(xbar, ybar) = (My/M , Mx/M)

then we just need to integrate each expression separately...

1. My = ∫[0 to 1] (x [x^(1/5) - x^5]) dx = 24/77. -----> (24/77)/(2/3) = 36/77

Mx = 1/2 ∫[0 to 1] [x^2/5 - x^10] dx = 24/77. ------> (24/77)/(2/3) = 36/77

M = ∫[0 to 1] (x^(1/5) - x^5) dx = 2/3

Centroid (36/77 , 36/77)