Centroids by Integration

1
Centroids by Integration

• Recall

2
Reduction to Single Integral

• Element area dA ydx
• Element centroid
• xel x yel y/2

3
Another way

• Element area dA (a-x)dy
• Element centroid
• xel x(a-x)/2 yel y

4
Polar Coordinates

• Element area dAr2dq/2
• Element centroid rel2r/3

5
Theorems of Pappus and Guldinus

• A surface of revolution is a formed by rotating a
  plane curve about an axis.

6
more

• A solid of revolution is a formed by rotating a
  plane area about an axis.

7
Theorem 1

• A 2p L

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Theorem 2

• V 2p A