Final answer:
The centroid of the region bounded by y = e^x, y = 0, x = 0, x = 5 is (2, 2e).
Step-by-step explanation:
The centroid of a region bounded by curves can be found by using the formula for the x-coordinate of the centroid:
x = (1/A) ∫[a,b] x f(x) dx
In this case, the region is bounded by y = e^x, y = 0, x = 0, and x = 5.
To find the centroid, we need to find the area of the region, A, and the integral of x times the curve function, f(x).
The area can be found by:
A = ∫[a,b] f(x) dx
For this region, the area is:
A = ∫[0,5] e^x dx
The integral of x times the curve function is:
∫[a,b] x f(x) dx = ∫[0,5] x e^x dx
Using these values, we can find the x-coordinate of the centroid:
x= (1/A) ∫[0,5] x e^x dx
After calculating the values of A and we get the centroid coordinates as (2, 2e).