Find the coordinates of the centroid of the following volume of revolution formed by rotating the area bounded by y^2-4=-x^2, x=0,x=1,
And the "x" axis about the "x" axis.
To find the coordinates of the centroid of a volume of revolution, we need to follow a few steps:
1. Find the equation of the curve: Start by rearranging the given equation, y^2 - 4 = -x^2, to get y^2 = 4 - x^2.
2. Find the limits of integration: Since we are rotating the area bounded by y^2 - 4 = -x^2, x = 0, and x = 1 about the x-axis, we need to find the limits of integration. For this case, the limits are x = 0 and x = 1.
3. Set up the integral: The method of finding the centroid of a volume of revolution involves integrating the cross-sectional area multiplied by its respective distance from the axis of rotation. In this case, we are rotating about the x-axis, so the distance from a general point (x, y) to the x-axis is y.
The formula for the centroid is given by: x_c = (1/A) * ∫x * dA and y_c = (1/A) * ∫y * dA,
where x_c and y_c are the coordinates of the centroid, A is the area of the region bounded by the given curve, and dA is the differential area.
4. Calculate the area: Use the formula A = ∫y * dx, where y = √(4 - x^2), and integrate it over the given limits, which are x = 0 and x = 1. This will give you the total area.
5. Calculate the integrals: Evaluate ∫x * y * dx and ∫y * y * dx over the given limits, which are x = 0 and x = 1.
6. Calculate the centroid coordinates: Divide the integrals from step 5 by the area obtained in step 4. The resulting values will be the x-coordinate (x_c) and y-coordinate (y_c) of the centroid.
By following these steps, you should be able to find the coordinates of the centroid of the volume of revolution formed by rotating the given area bounded by y^2 - 4 = -x^2, x = 0, x = 1 about the x-axis.