centroid by integration

709 Centroid of the area bounded by one arc of sine curve and the x-axis

Problem 709
Locate the centroid of the area bounded by the x-axis and the sine curve $y = a \sin \dfrac{\pi x}{L}$ from x = 0 to x = L.
 

708 Centroid and area of spandrel by integration

Problem 708
Compute the area of the spandrel in Fig. P-708 bounded by the x-axis, the line x = b, and the curve y = kxn where n ≥ 0. What is the location of its centroid from the line x = b? Determine also the y coordinate of the centroid.
 

Centroid and area of spandrel under the curve y = kx^n

 

707 Centroid of quarter ellipse by integration

Problem 707
Determine the centroid of the quadrant of the ellipse shown in Fig. P-707. The equation of the ellipse is $\dfrac{x^2}{a^2} + \dfrac{y^2}{b^2} = 1$.
 

Centroid of quarter ellipse in the first quadrant

 

706 Centroid of quarter circle by integration

Problem 706
Determine the centroid of the quarter circle shown in Fig. P-706 whose radius is r.
 

Centroid of quarter circle

 

705 Centroid of parabolic segment by integration

Problem 705
Determine the centroid of the shaded area shown in Fig. P-705, which is bounded by the x-axis, the line x = a and the parabola y2 = kx.
 

Open to the right parabola in the first quadrant

 

Centroids and Centers of Gravity

Centroids of Composite Figures

Center of gravity of a homogeneous flat plate
$W \, \bar{x} = \Sigma wx$

$W \, \bar{y} = \Sigma wy$

 

Centroids of areas
$A \, \bar{x} = \Sigma ax$

$A \, \bar{y} = \Sigma ay$

 

Centroids of lines
$L \, \bar{x} = \Sigma lx$

$L \, \bar{y} = \Sigma ly$

 

Center of Gravity of Bodies and Centroids of Volumes

Center of gravity of bodies
$W \, \bar{x} = \Sigma wx$

$W \, \bar{y} = \Sigma wy$

$W \, \bar{z} = \Sigma wz$

 

Centroids of volumes
$V \, \bar{x} = \Sigma vx$

$V \, \bar{y} = \Sigma vy$

$V \, \bar{z} = \Sigma vz$

 

 
 
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