Struggling with the Shell Method in Calculus?

[mathnoobie]

Homework Statement

So I'm completely confused on how to solve shell method problems. I think understand it, and then there's a problem that shows that I do not understand it at all. So I want to start basic

When you're the following :
rotated about the x-axis
#1)

y=x^3
x=0
y=8

2)x+y=4
y=x
y=0

Homework Equations

2∏∫p(y)h(y)dy

The Attempt at a Solution

1) So I solve for x and get x=cubert(y)
Now what?
I want to know why I would plug in Y for the radius, and how do I find the limits of integration from this? It seems that sometimes you just plug in Y and sometimes it's some long equation.

2) Again, I solve for X, so h(x)=4-y and once again am stumped on what to do next, how do I know what p(x) is and what the limits of integration are. I don't know what purpose y=x serves and what y=0 serves.

 

[Mark44]

Homework Statement

So I'm completely confused on how to solve shell method problems. I think understand it, and then there's a problem that shows that I do not understand it at all. So I want to start basic

When you're the following :
rotated about the x-axis
#1)

y=x^3
x=0
y=8

2)x+y=4
y=x
y=0

Homework Equations

2∏∫p(y)h(y)dy

The Attempt at a Solution

1) So I solve for x and get x=cubert(y)
Now what?
I want to know why I would plug in Y for the radius, and how do I find the limits of integration from this? It seems that sometimes you just plug in Y and sometimes it's some long equation.

Did you sketch a graph of the function, and another of the solid of revolution? Students often skip these steps, thinking that they are extra work, but they usually make the difference between being able to work the problem and getting hopelessly lost.

For this problem, I would be more inclined to use disks - do you have to use shells?

To answer your question, you don't just "plug in" y in this problem. As you can see from your sketch (you have one, right?) the radius is the distance from the line y = 8 to the curve.

2) Again, I solve for X, so h(x)=4-y and once again am stumped on what to do next, how do I know what p(x) is and what the limits of integration are. I don't know what purpose y=x serves and what y=0 serves.

 

[mathnoobie]

Yes, unfortunately I have to use the shell method.
And I graphed #1 on my calculator(I always graph, problem is I don't understand everything it means when it comes to the shell method), but it doesn't help me understand what happens and why.

I missed class the day of the shell method and I can't seem to grasp it just by examining the book.

 

[Mark44]

The shell method can be thought of as calculating the volume of a typical element, and then adding all those bits up to get the total volume. It's sort of like taking an onion apart.

I'm not sure that graphing on a calculator is the best way to go about it, since the calculator can't draw in the shells or disks or whatever. I would advise using the calculator to get the basic graph, and then sketching a more useful graph on paper.

For this problem (#1), using the shells method, you have an area element whose dimensions are Δy by x, so its area is x Δy. This area is revolved around the line y = 8, and that sweeps out a "shell". Sketch this shell and find the radius that is used in the formula you showed.

 

[mathnoobie]

Ah, I understand now. Thank you