How do you use the disk or shell method to find the volume of the solid generated by revolving the regions bounded by the graphs of #y = x^(1/2)#, #y = 2#, and #x = 0# about the line #x = -1#?

Question

Elijah Abram · Accepted Answer

See the explanation section, below.

Here is a picture of the region with a thin slice taken vertically.

So thickness is #dx# and we will use cylindrical shells.

The representative shell will have volume
#2pi "radius" * "height" * "thickness" = 2pi(x+1)(2-sqrtx)dx#.

We see that the values of #x# vary from #0# to #4#

So the volume of the solid is found by evaluating:

#int_0^4 2pi(x+1)(2-sqrtx)dx = 2pi int_0^4 (x+1)(2-sqrtx)dx #

Expand the product and integrate term by term.

To use disks/washers , we need to take our slices perpendicular to the axis of rotation as shown below.

The curve is now expressed as #x=y^2#

The representative washer has volume: #pi(R^2-r^2)dy# Where #r# is the greater radius (#x=-1# to #x=y^2#) and #r# is the lesser radius (#x=-1# to #x=0#) and the thickness is #dy#.

#y# goes from #0# to #2#, so the Volume is

#int_0^2 pi((y^2+1)^2-(1)^1) dy = pi int_0^2 ((y^2+1)^2-(1)^1) dy#

Again, expand and integrate term by term.

Isabella Ackerman · Answer

undefined To find the volume of the solid generated by revolving the region bounded by the graphs of $ y = \sqrt{x} $, $ y = 2 $, and $ x = 0 $ about the line $ x = -1 $, we can use the disk or shell method.

Disk Method:
1. Determine the limits of integration. Since we are revolving about the line $ x = -1 $, the limits of integration will be from $ x = 0 $ to $ x = 3 $ (the intersection point of $ y = \sqrt{x} $ and $ y = 2 $).
2. Express the radius of the disks as the distance from the line of revolution ($ x = -1 $) to the curve. In this case, the radius $ r $ is $ 1 + x $ (since $ x = -1 $ is one unit to the left of the axis of revolution).
3. Write the volume of each disk as $ \pi r^2 \Delta x $.
4. Integrate the expression $ \pi r^2 \Delta x $ from the lower limit to the upper limit to find the total volume.

Shell Method:
1. Determine the limits of integration as before.
2. Express the radius of the shells as the distance from the axis of revolution ($ x = -1 $) to the curve. In this case, the radius $ r $ is $ x + 1 $.
3. Express the height of the shells as the difference in $ y $ values of the bounding curves. Here, the height $ h $ is $ 2 - \sqrt{x} $.
4. Write the volume of each shell as $ 2\pi r h \Delta x $ (since the shells are formed by revolving about a vertical line).
5. Integrate the expression $ 2\pi r h \Delta x $ from the lower limit to the upper limit to find the total volume.

Both methods will yield the same result, but the choice between them depends on which one is easier to set up and integrate in a given scenario.