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Chapter 5: Q24MP (page 275)

Repeat Problem 23a for the paraboloidzc=x2a2+y2b20<z<c.

Short Answer

Expert verified

The centroid of the solid paraboloid is x,y,z=0,0,23c

Step by step solution

01

 Step 1: Given Condition

The paraboloid is zc=x2a2+y2b2.0<z<c

02

Concept of centroid

The centroid or geometric center of a plane figure is the arithmetic mean position of all the points in the figure.

03

 Step 3: Calculate the volume

The volume is given by the following formula,

V=鈭�x鈭�y鈭�zdxdydz=鈭�x鈭�y鈭�zdx'dy'dz'abc=abc鈭�vdV=2蟺补产肠鈭�01dz'12z'=蟺2abc

Thus,

x=y=0
04

Calculate the  z coordinate

Now, to calculate the value of z ,

V=鈭�x鈭�y鈭�zzdxdydz=abc2鈭�x鈭�y鈭�zdx'dy'dz'z'=abc2鈭�02蟺d胃鈭�01dz'z鈭�0zrdr=2蟺补产肠鈭�01z'dz'12z'=蟺补产肠2鈭�01z'2dz'=蟺3cz=23c

Hence, the centroid is .x,y,z=0,0,23c.

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Most popular questions from this chapter

Above the rectangle with vertices (0,0), (0,1),(2,0),(2,1), and below the surface z2=36x2(4-x2)

A thin rod 10 ft long has a density which varies uniformly from 4 to 24 lb/ft. Find

(a) M,

(b)x,

(c) lmabout an axis perpendicular to the rod,

(d)labout an axis perpendicular to the rod and passing through the heavy end.

鈭�鈭�ydxdyover the triangle with vertices (-1,0),(0,2),and(2,0)

Use Problems 12 and 13 to find the centroids of a semi-circular area and of a semi-circular arc. Hint: Assume the formulas A=4蟺r2, V=43蟺r3 for a sphere.

鈭�x=01鈭�y=03-3xdydx
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