Find the center of mass of the region bounded by the curve
y = x3 - 4x2 +3x ; the x-axis ; x = 0 ; x = 1
To find the center of mass of the region bounded by the curve y = x^3 - 4x^2 + 3x, the x-axis, x = 0, and x = 1, you will need to integrate the function to find the moments about the x and y axes, and then divide by the total area of the region.
Step 1: Find the area of the region.
To find the area of the region bounded by the curve, you need to integrate the function y = x^3 - 4x^2 + 3x from x = 0 to x = 1.
Area = ∫[0,1] (x^3 - 4x^2 + 3x) dx
Step 2: Find the moments about the x and y axes.
To find the moment about the x-axis, you need to integrate the function y = x^3 - 4x^2 + 3x multiplied by x, from x = 0 to x = 1.
M_x = ∫[0,1] (x)(x^3 - 4x^2 + 3x) dx
To find the moment about the y-axis, you need to integrate the function y = x^3 - 4x^2 + 3x multiplied by y = 0 (since the y-coordinate is 0 for points on the x-axis), from x = 0 to x = 1.
M_y = ∫[0,1] 0(x^3 - 4x^2 + 3x) dx
Step 3: Calculate the center of mass.
The center of mass coordinates (x_c, y_c) can be found using the following formulas:
x_c = M_y / Area
y_c = M_x / Area
Calculate the integrals for the area, M_x, and M_y in Step 1 and Step 2. Then, plug the values into the center of mass formulas in Step 3 to find the coordinates (x_c, y_c) of the center of mass.