Use the method of cylindrical shells to find the volume generated by rotating the region bounded by the given curves about the y-axis.

y = 4x^2, y = 24x − 8x^2

Question

Use the method of cylindrical shells to find the volume generated by rotating the region bounded by the given curves about the y-axis.

y = 4x^2,    y = 24x − 8x^2

psymon · Answer

Well, first set both equations equal to each other so we can see where they intersect.

anonymous · Answer

it's a webassign question . I worked on this question but my answer was wrong..!

psymon · Answer

Well, imjust saying the first step is to check their intersection. We need to know where they intersect so we can see what our integration limits will be.

anonymous · Answer

0,2 ?

psymon · Answer

Right, those would be our limits. Now since we're using shell method and the axis of revolution is vertical, we have this formula:
$$2 \pi \int\limits_{a}^{b}p(x)h(x)dx$$where p(x) is the radius and h(x) is the height. Now the height refers to the portion underneath the graphs. So a lot of the time the height turns out to be the graph itself as long as its bounded properly. So this is a visual of what we have:

|dw:1378617713233:dw|

So now just to get an idea of what height actually is, let's first pretend that we only had the function 24x-8x^2. Because the area bounded is UNDERNEATH the graph of 24x-8x^2, the height we use is the function itself:

|dw:1378617851984:dw|

Now if the function were only the 4x^2 part we would have this:

|dw:1378617929340:dw|

Problem is in our problem we only have the area in between the two functions. So in order to get the appropriate height of the graph, we subtract 4x^2 from 24x-8x^2

|dw:1378618025419:dw|

So this makes out height = 24x-8x^2 -4x^2 = 24x-12x^2

Now the radius is the distance from the axis of revolution to the graph itself. When the axis of revolution is only the y-axis, we say the radius is "x". Now if the radius were something other than either of the major axes, we would have to do some sort of addition or subtraction, but we do not have that in this case. So since we have the y-axis as our axis of revolution, our radius is x, meaning the integral we have is:

$$2\pi \int\limits_{0}^{2}(x)(24x-12x ^{2})dx \implies 2\pi \int\limits_{0}^{2}(24x ^{2}-12x ^{3})dx$$

I know thats a lot of info, so check it out first o.o