Use integration to find the center of mass of the triangle (0,0) (2,0) (0,8).

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justaguide eNotes educator| Certified Educator

The center of mass of the the triangle with vertex (0, 0), (2, 0) and (0, 8) has to be determined.

The equation of the line passing through (0, 0) and (2, 0) is y = 0. The equation of the line passing through (0,0) and (0, 8) is x = 0 and the equation of the line passing through (2, 0) and (0, 8) is (y - 0)/(x - 2) = (8 - 0)/(0 - 2) = 8/-2 = -4

=> y = 8 - 4x

The diagram of the triangle is:

The center of mass of the triangle is given by:

`M_x = (int_0^2 (x*(8 - 4x))dx)/(int_0^2 8 - 4x dx)`

=> `(int_0^2 8x - 4x^2 dx)/(int_0^2 8 - 4x dx)`

=> `((8x^2/2 - 4x^3/3)_0^2)/((8x - 2x^2)_0^2)`

=> `((4x^2 - 4x^3/3)_0^2)/8`

=> `(16 - (4/3)*2^3)/8`

=> `2/3`

`M_y = (int_0^2 (1/2)(8 - 4x)^2 dx)/(int_0^2 8 - 4x dx)`

=> `(int_0^2 (1/2)(64 - 64x + 16x^2) dx)/(int_0^2 8 - 4x dx)`

=> `(int_0^2 32 - 32x + 8x^2 dx)/(int_0^2 8 - 4x dx)`

=> `(32x - 16x^2 + 8x^3/3)_0^2 / 8`

=> `8/3`

The center of mass of the triangle is `(2/3, 8/3)`