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 = 5x 2. Use slices perpendicular to the y-axis. Each slice has balance point: Bounds: To matching answers for M y (with length) use the property: (a - b)(a + b) = a 2 - b 2

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 = x 2 and y = x + 2. Use slices perpendicular to the x-axis. Each slice has balance point: Bounds: Top: Bottom: To matching answers for M x (with length) use the property: (a - b)(a + b) = a 2 - b 2

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 Parts Back to: Choose u (to differentiate (“du”)) dv (to integrate (“v”)) This second integral is simpler than the first

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

10. Integration by Parts Integrate: