## 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.

**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 = kx^{n} where n ≥ 0. What is the location of its centroid from the line x = b? Determine also the y coordinate of the centroid.

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**Problem 707**

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

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**Problem 706**

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

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**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 y^{2} = kx.

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Centroids and Centers of Gravity
## Centroids of Composite Figures

Center of gravity of a homogeneous flat plate
Centroids of areas
Centroids of lines
## Center of Gravity of Bodies and Centroids of Volumes

Center of gravity of bodies
Centroids of volumes

$W \, \bar{x} = \Sigma wx$

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

$A \, \bar{x} = \Sigma ax$

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

$L \, \bar{x} = \Sigma lx$

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

$W \, \bar{x} = \Sigma wx$

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

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

$V \, \bar{x} = \Sigma vx$

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

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

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