find the centroid of the area of the finite region enclosed by the curve y=x^(2)+1 when x-axis and the line x=0 and x=3
To find the centroid of a finite region enclosed by a curve, we need to calculate the coordinates of the centroid using the formula:
x-coordinate of centroid = (1/area) * ∫(x * y) dx
y-coordinate of centroid = (1/2 * area) * ∫(y^2) dx
Step 1: Find the area of the region enclosed by the curve.
To find the area, we need to integrate the curve equation with respect to x, within the given limits (from x = 0 to x = 3).
Area = ∫(x^2 + 1) dx, limits: 0 to 3
Calculating this integral will give us the area of the region.
Step 2: Calculate the x-coordinate of the centroid.
Using the formula stated earlier, we need to evaluate the integral of (x * y) with respect to x:
x-coordinate of centroid = (1/area) * ∫(x * (x^2 + 1)) dx, limits: 0 to 3
This will give us the x-coordinate of the centroid.
Step 3: Calculate the y-coordinate of the centroid.
Using the formula stated earlier, we need to evaluate the integral of (y^2) with respect to x:
y-coordinate of centroid = (1/(2 * area)) * ∫((x^2 + 1)^2) dx, limits: 0 to 3
This will give us the y-coordinate of the centroid.
Step 4: Evaluate the integrals and compute the centroid coordinates.
Calculate the integrals in Step 2 and Step 3 to obtain the x-coordinate and y-coordinate of the centroid, respectively. Finally, write the centroid as an ordered pair (x, y).
Please note that to perform the integrations and computations, you may need to use integration techniques and basic calculus operations.