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To get the volume and the
To get the volume and the centroid of the solid generated when the area bounded by the x-axis, line x=1 and the curve y=x2 is rotated about the x and y-axis, here it is:
To get the volume of the solid generated when the resulting area is rotated about x-axis, we need a figure. It looks like this:
We can imagine that we could create a solid figure by using very small cylinders, so we will use the circular disk method to get the whole volume.The formula used for the circular disk method is...
V=π∫bay2dx
Basing from the figure shown above, the equation we need is:
V=π∫bay2dx
Then...
V=π∫10y2dx V=π∫10(x2)2dx V=π∫10x4dx V=π∫10x4dx V=15π
The volume generated when the area is rotated around x−axis is 15π cubic units
To get the centroid of the solid generated when the resulting area is rotated about x-axis, we will do this:
The general equation to get the centroid of a figure is...
Vˉx=∫xcdV
where V is the volume of the solid, ˉx is the coordinate of the centroid of the resulting solid, and dV is the element of volume, usually a disk, ring or shell. Because of symmetry of the solid that lies on its axis, we only need one coordinate to find its centroid.
In this case, the equation we need to get the centroid is:
Vˉx=∫xcdV
Now solving for the centroid:
Vˉx=∫xcdV (15π)ˉx=∫10x(πy2dx) (15π)ˉx=∫10x(π(x2)2dx) (15π)ˉx=∫10πx5dx (15π)ˉx=16π xc=56
Therefore, the centroid of the volume that lies on x - axis is xc=56
To get the volume of the solid generated when the resulting area is rotated about y-axis, we need a figure. It looks like this:
We can imagine that we could create a solid figure by using very small cylindrical shells, so we will use the cylindrical shell method to get the whole volume.
The formula used for the circular disk method is...
V=∫ba2πxydx
The volume of the cylindrical shell is expressed as circumference of circle × height × thickness or...
The volume of the cylindrical shell is expressed as 2πr × h × t or in this problem's context...
The volume of the cylindrical shell element dV is expressed as 2πx × y × dx.
Basing from the figure shown above, the equation we need is:
V=∫ba2πxy2dx
Then solving for its volume:
V=∫ba2πxydx V=∫102πx(x2)dx V=∫102πx(x4)dx V=∫102πx5dx V=12π
The volume generated when the area is rotated around y - axis is 12π cubic units
To get the centroid of the solid generated when the resulting area is rotated about y-axis, we will do this:
The general equation to get the centroid of a figure is...
Vˉx=∫xcdV
where V is the volume of the solid, ˉx is the coordinate of the centroid of the resulting solid, and dV is the element of volume, usually a disk, ring or shell. Because of symmetry of the solid that lies on its axis, we only need one coordinate to find its centroid.
In this case, the equation we need to get the centroid is:
Vˉy=∫ycdV
Now solving for the centroid:
Vˉy=∫ycdV (12π)ˉy=∫1012y(2πxydx) (12π)ˉy=∫10yπxydx (12π)ˉy=∫10πxy2dx (12π)ˉy=∫10πx(x2)2dx (12π)ˉy=∫10πx5dx (12π)ˉy=16π yc=13
Therefore, the centroid of the volume that lies on y - axis is yc=13
Sana makakatulong...:-)