Integral calculus: volume and centroid of the solid

Please help po dito

Region 1 is bounded by the x axis, the line x=1 and the curve y=x^2.find the volume and centroid of the solid generated when the indicated region is revolved about y axis and revolved about x axis

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:

pic1.png

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=πbay2dx

Basing from the figure shown above, the equation we need is:

V=πbay2dx

Then...

V=π10y2dx V=π10(x2)2dx V=π10x4dx V=π10x4dx V=15π

The volume generated when the area is rotated around xaxis 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:

pic2.png

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...:-)