Volume by integration

Find the volume and centroid of the solid generated using disc and shel methods,as the region bounded by the curve y^2=8x, the x axis and the latus rectun of the curve is revolved about:
A. The x axis
B. The y axis
C. The latus rectum

A.) To get the volume and centroid of the solid of revolution generated by the curve y2=8x, the x-axis and the latus rectum of y2=8x when its area rotates about the x-axis, the solution look like this:

We need a figure. It looks like this:

picture1.png

Adding some finer details, it looks like this:

picture2.png

The chord that passes through the parabola and perpendicular to the axis is the latus rectum of a parabola and its length is 4a if the equation of parabola is y2=4ax as shown below.

picture3.png

The volume of solid of revolution generated by the curve y2=8x, the x-axis and the latus rectum of y2=8x when its area rotates about the x-axis can be get by the disk method.

One disk element has the volume dV and its volume is πr2dh, where dV is the volume element, r is the radius and dh is the height element. In this case:

picture4.png

Now getting the volume of the figure above:

For one volume element for the figure above, its volume is: dV=πr2dh

If we are going to add many volume elements to create a solid figure, the volume becomes:

V=πr2dh

In this case, the volume of the solid generated above is:

V=πy2dx

Now getting its volume:

V=πy2dx V=π(8x)dx V=20π(8x)dx V=16π cubic units

Now getting its centroid; the centroid of the figure above can be seen on the x-axis because of symmetry as shown below:

picture5.png

Because of figure's symmetry, there is only one coordinate where the centroid of the figure lies. In this case, the centroid lies on the x-axis. The formula to get the centroid of the figure is:

Vˉx=xcdV

Then, in this case, the centroid of the figure above is:

16πˉx=xdV

Then:

16πˉx=x(πy2dx) 16πˉx=20x(π(8x)dx) 16πˉx=208πx2dx 16πˉx=643π ˉx=43

picture5_0.png

B.) To get the volume and centroid of the solid of revolution generated by the curve y2=8x, the x-axis and the latus rectum of y2=8x when its area rotates about the y-axis, the solution look like this:

We need a figure. It looks like this:

picture1.png

Adding some finer details, it looks like this:

picture2.png

The chord that passes through the parabola and perpendicular to the axis is the latus rectum of a parabola and its length is 4a if the equation of parabola is y2=4ax as shown below.

picture3.png

The volume of solid of revolution generated by the curve y2=8x, the x-axis and the latus rectum of y2=8x when its area rotates about the y-axis can be get by the ring method.

To get the volume of the ring, subtract volume of one big ring by volume of one small ring as shown below:

picture9.png

One ring element has the volume dV and its volume is π(r22r21)dh, where dV is the volume element, r2 is the radius of big cylinder, r21 is the radius of one small cylinder and dh is the height element. In this particular problem:

picture10.png

Now getting the volume of the figure above:

For one volume element for the figure above, its volume is:
dV=π(r22r21)dh

If we are going to add many volume elements to create a soid figure, the volume becomes:

V=π(r22r21)dh

In this case, the volume of the solid generated above is:

V=π((2)2(x)2)dy

Now getting its volume:

V=40π((2)2(x)2)dy V=40π(4x2)dy V=40π(4(y464))dy V=645π cubic units

Now getting its centroid; the centroid of the figure above can be seen on the y-axis because of symmetry as shown below:

picture11.png

Because of figure's symmetry, there is only one coordinate where the centroid of the figure lies. In this case, the centroid lies on the y-axis. The formula to get the centroid of the figure is:

Vˉy=ycdV

Then, in this case, the centroid of the figure above is:

645πˉy=ydV

Then looking at the picture above, its obvious that the centroid of the figure below is yc=0

C.) To get the volume and centroid of the solid of revolution generated by the curve y2=8x, the x-axis and the latus rectum of y2=8x when its area rotates about the line x=2 (the equation of the latus rectum of the curve y2=8x), the solution look like this:

We need a figure. It looks like this:

picture12.png

The volume of solid of revolution generated by the curve y2=8x, the x-axis and the latus rectum of y2=8x when its area rotates about the x=2 can be get by the ring method.

To get the volume of the ring, subtract volume of one big ring by volume of one small ring as shown below:

picture9.png

One ring element has the volume dV and its volume is π(r22r21)dh, where dV is the volume element, r2 is the radius of big cylinder, r21 is the radius of one small cylinder and dh is the height element. In this particular problem:

picture12.png

Now getting the volume of the figure above:

For one volume element for the figure above, its volume is:
dV=π(r22r21)dh

If we are going to add many volume elements to create a soid figure, the volume becomes:

V=π(r22r21)dh

In this case, the volume of the solid generated above is:

V=π((2)2(x)2)dy

Now getting its volume:

V=44π((2)2(x)2)dy V=44π(4x2)dy V=44π(4(y464))dy V=1285π cubic units

Now getting its centroid; the centroid of the figure above can be seen on the x-axis because of symmetry as shown below:

picture12.png

Because of figure's symmetry, there is only one coordinate where the centroid of the figure lies. In this case, the centroid lies on the x-axis. The formula to get the centroid of the figure is:

Vˉy=ycdV

Then, in this case, the centroid of the figure above is:

1285πˉy=xdV

Then looking at the picture above, its obvious that the centroid of the figure below is (x,y)=(2,0)

Alternate solutions are encouraged.....