1. Centers of Mass Review & Integration by Parts
Chapter 7.1
March 20, 2007

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:

4. Matching Answers: Using the property: (a - b)(a + b) = a2 - b2
Becomes:

5. 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:

6. Again to match answers: (a + b)(a - b) = a2 - b2
Becomes:

7. 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.

8. 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:

9. 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)

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

11. Checking the answer:

12. Integration by Parts Examples:

13. Integration by Parts Examples: