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Centers of Mass Review & Integration by Parts

Centers of Mass Review & Integration by Parts. Center of Mass: 2-Dimensional Case. The System’s Center of Mass is defined to be:.

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Centers of Mass Review & Integration by Parts

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  1. Centers of Mass Review &Integration by Parts

  2. Center of Mass: 2-Dimensional Case • The System’s Center of Mass is defined to be:

  3. Find the center of mass of the the lamina R with density 1/2 in the region in the xy plane bounded by y = 6x -1 and y = 5x2. Use slices perpendicular to the y-axis. Bounds: Each slice has balance point: To matching answers for My (with length) use the property: (a - b)(a + b) = a2 - b2

  4. Find the center of mass of the the lamina R with density 1/3 in the region in the xy plane bounded by y = x2 and y = x + 2. Use slices perpendicular to the x-axis. Bounds: Top: Bottom: Each slice has balance point: To matching answers for Mx (with length) use the property: (a - b)(a + b) = a2 - b2

  5. Integration by Parts: “Undoing” the Product Rule for Derivatives • Consider: • We have no formula for this integral. • Notice that x and ln(x) are not related by derivatives, so we cannot use the substitution method.

  6. Integration by Parts: “Undoing” the Product Rule for Derivatives • Look at the derivative of a product of functions: • Let’s use the differential form: • And solve for udv • Integrating both sides, we get:

  7. Integration by Parts: “Undoing” the Product Rule for Derivatives • Integrating both sides, we get: • Or • The integral should be simpler that the original • If two functions are not related by derivatives (substitution does not apply), choose one function to be the u (to differentiate) and the other function to be the dv (to integrate)

  8. Integration by PartsBack to: • Choose u (to differentiate (“du”))dv (to integrate (“v”)) This second integral is simpler than the first

  9. Integration by PartsEvaluate: • Choose u (to differentiate (“du”))dv (to integrate (“v”))

  10. Integration by PartsIntegrate:

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