Solid Revolution Cylindrical Shell Method

[burt]

What is the volume of the solid of revolution formed by shifting the region bounded by the curve [MATH]y=x^2[/MATH], the line [MATH]y=0[/MATH], and the line [MATH]x=1[/MATH] over by one unit along the positive x‑axis and revolving the resulting region around the xx‑axis?

Use the cylindrical shell method.

I can't seem to figure this one out. What is the point of shifting the graph over one? It doesn't seem to make a difference when I picture the solid...
This is what I got: [MATH]2\pi\int^1_0(2-\sqrt{y+1})(y)\ dy\\=2\pi\int^1_0(2y-y\sqrt{y+1})\ dy\\=2\pi y-\left(\frac{1y(y+1)^\frac32}{3}-\frac{2(y+1)^\frac52}{5}\right)\\=\frac{5\pi}3[/MATH]The only thing is that I don't think this is right...

 

[Romsek]

shifting the whole thing over 1 unit would result in new boundaries

\(\displaystyle y=(x-1)^2,~x=2,~y=0\)

Then this is rotated about the y axis.

 

[burt]

shifting the whole thing over 1 unit would result in new boundaries

\(\displaystyle y=(x-1)^2,~x=2,~y=0\)

Then this is rotated about the y axis.

True, but the amount of space is the same - so why does it matter?

 

[Romsek]

The cross sectional area remains the same but it is revolved with a larger radius.

 

[burt]

t is revolved around a larger radius.

Why is this? It is being revolved around the x-axis, not the y-axis?

 

[burt]

The cross sectional area remains the same but it is revolved with a larger radius.

Perhaps what I'm thinking is wrong. I think this is the shifted solid:


And this is the original:


They look the same to me...

 

[Romsek]

Why is this? It is being revolved around the x-axis, not the y-axis?

my apologies. you are correct. translation along the axis of rotation has no affect on the volume.

 

[Romsek]

\(\displaystyle V=\displaystyle \int_1^2\pi( (x-1)^2)^2~dx = \\

\displaystyle \pi \int_0^1 x^4~dx = \dfrac \pi 5
\)

 

[burt]

\(\displaystyle V=\displaystyle \int_1^2\pi( (x-1)^2)^2~dx = \\

\displaystyle \pi \int_0^1 x^4~dx = \dfrac \pi 5
\)

I see you put n the x-1. Did you need to? Could you have done [MATH]x^2[/MATH]?

 

[Romsek]

I see you put n the x-1. Did you need to? Could you have done [MATH]x^2[/MATH]?

I could have done \(\displaystyle x^2\) with the integration limits 0 to 1. That's what the second line is.

Problem says shift it over so that's where I started.