15 x L C x L C x L C x x x x x M A y y y M y y dx M y y dx x x dx M x x dx x x

15 x l c x l c x l c x x x x x m a y y y m y y dx m y

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15 x L C x L C x L C x x x x x M A y y y M y y dx M y y dx x x dx M x x dx x x M M M M cu units
1 15 1 6 2 5 x x M A y M y A y units 1 12 1 6 1 2 y y M A x M x A x units 1 2 , 2 5 C
2 x y 2 8 x y   Example Determine the centroid of the area bounded by the parabolas and . 2 x y 0,0 V Curve 1: dy y x (x 1 ,y) (x 2 ,y) (4,-2) 2 x y 2 8 x y   2 1 x x x , C x y y Curve 2: 2 8 x y   0,0 V Solving for intersection points: 2 2 4 4 3 8 8 8 0 8 0 0; 2 0; 4 y y y y y y y y y y x x       Therefore, the intersection points are (0, 0) and (4, -2).
   3 2 3 2 2 1 0 2 1 2 2 2 2 1 0 2 2 0 3 0 2 2 0 3 2 3 8 8 8 8 2 2 2 2 3 2 1 1 3 3 2 2 1 2 2 2 3 3 2 8 8 3 3 16 8 3 3 8 3 dA x x dy A x x dy x x y x y x y x x y A y y dy y A y dy y A y A A A A         : : . sq units Solving for area A: Continue solving for the centroid
Find the centroid of each of the given plane region bounded by the following curves: 1. y = 10x x 2 , the x-axis and the lines x = 2 and x = 5 2. 2x + y = 6, the coordinate axes 3. y = 2x + 1, x + y = 7, x = 8 4. y 2 = 2x, y = x 4 5. y = x 3 , y = 4x [first quadrant] 6. y 2 = x 3 , y = 2x 7. y = x 2 4, y = 2x x 2 8. the first quadrant area of the circle x 2 +y 2 = a 2 9. the region enclosed by b 2 x 2 + a 2 y 2 = a 2 b 2 in the first quadrant
TOPIC APPLICATIONS CENTRIODS OF SOLIDS OF REVOLUTION
Center of gravity of a solid of revolution The coordinates of the centre of gravity of a solid of revolution are obtained by taking the moment of an elementary disc about the coordinate axis and then summing over all such discs. Each sum is then approximately equal to the moment of the total volume taken as acting at the centre of gravity. Again, as the disc thickness approaches zero the sums become integrals: 2 2 and 0 b x a b x a xy dx x y y dx
THE MOMENT OF A SOLID of volume V, generated by revolving a plane area about a horizontal or vertical axis, with respect to the plane through the origin and perpendicular to the axis may be found as follows: 1.Sketch the region, showing a representative strip and the approximating rectangle. 2.Form the product of the volume, disc or shell generated by revolving the rectangle about the axis and the distance of the centroid of the rectangle from the plane, and sum for all the rectangles. 3.Assume the number of rectangles to be indefinitely increased and apply the fundamental theorem.
When the area is revolved about the x-axis, the centroid is on that axis. Which means that, for solids generated by revolving the plane area about an axis, its centroid is on that axis, thus, giving one coordinate.

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