Centers of Mass and Centroids

Find the centroid of a thin, flat plate covering the region enclosed by the parabolas 𝔂 = 2𝓍² and 𝔂 = 3 ― 𝓍² .

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Identify the region bounded by the curves \( y = 2x^2 \) and \( y = 3 - x^2 \). The centroid \( (\bar{x}, \bar{y}) \) of the region can be found by first determining the points of intersection of these two curves.

Set the two functions equal to find the intersection points: \( 2x^2 = 3 - x^2 \). Solve for \( x \) to find the limits of integration for the region.

Calculate the area \( A \) of the region by integrating the difference between the upper curve and the lower curve over the interval found: \[ A = \int_{a}^{b} \left( (3 - x^2) - (2x^2) \right) \, dx = \int_{a}^{b} (3 - 3x^2) \, dx \].

Find the coordinates of the centroid using the formulas: \[ \bar{x} = \frac{1}{A} \int_{a}^{b} x \left( (3 - x^2) - (2x^2) \right) \, dx = \frac{1}{A} \int_{a}^{b} x (3 - 3x^2) \, dx \] and \[ \bar{y} = \frac{1}{2A} \int_{a}^{b} \left( (3 - x^2)^2 - (2x^2)^2 \right) \, dx \].

Evaluate the integrals and simplify the expressions to find \( \bar{x} \) and \( \bar{y} \), which give the coordinates of the centroid of the plate.

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주요 개념

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Centroid of a Region

The centroid is the geometric center or average position of all points in a plane figure. For a thin, flat plate with uniform density, it corresponds to the point where the plate would balance perfectly. It is found by calculating the average x and y coordinates weighted by the area.

Area Between Curves

To find the centroid, you first need the area of the region bounded by the given curves. This involves integrating the difference between the upper and lower functions over the interval where they intersect, which gives the total area of the enclosed region.

Moments About the Axes

The coordinates of the centroid are found using moments: the moment about the y-axis gives the x-coordinate, and the moment about the x-axis gives the y-coordinate. These moments are integrals of x or y multiplied by the area element, divided by the total area.

Centers of Mass and Centroids Find the centroid of a thin, flat plate covering the region enclosed by the parabolas 𝔂 = 2𝓍² and 𝔂 = 3 ― 𝓍² .