centroids

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.

Solution 705
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At (a, b)

Thus, → equation of parabola

Differential area

Area of parabola by integration

Location of centroid from the y-axis (x-intercept of centroid)

answer Location of centroid from the x-axis (y-intercept of centroid)

answer

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

Solution 706
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Equation of the circle with center at the origin and radius r is

Differential area

Area of the quarter circle

x-coordinate of the centroid

answer y-coordinate of the centroid

answer

Problem 707
Determine the centroid of the quadrant of the ellipse shown in Fig. P-707. The equation of the ellipse is .

Solution 707
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Equation of ellipse in y as a function of x

Differential area

Area of quarter ellipse

x-coordinate of the centroid

answer y-coordinate of the centroid

answer

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.

Solution 708
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At (b, h)

Thus,

Differential area

Area by integration

answer

x-coordinate of the centroid

Location of centroid from the line x = b

answer y-coordinate of