Volumes of solids of revolution

[Andiio]

I'm getting a bit confused over calculating volumes of revolution with a rotation about the y-axis.. like with what bounds (upper/lower) to choose and all; is the general gist just to find the different bounds that enclose the regions/areas the curves form with the y-axis and then calculating the volume through that?

Thanks!

[taiga]

What I do is cheat and make x a function of y. So you get x=arcsin(y) for example. Then make your horizontal axis y and your vertical axis x. Then find the volume as you usually would

[Andiio]

Ahhh okay haha I get what you mean.

But how would you calculate the volume (rotated about y) for a question like this: y = √x with the lines x=1 and x=4?

I've been stumped on this question for the past half an hour, I had a looked at the worked sols but don't understand why they did some of the working :\

[stonecold]

Ahhh okay haha I get what you mean.

But how would you calculate the volume (rotated about y) for a question like this: y = √x with the lines x=1 and x=4?

I've been stumped on this question for the past half an hour, I had a looked at the worked sols but don't understand why they did some of the working :\

You just find the corresponding y values, then sub them in, so in this case, it would be between y=1 and y=2

[Andiio]

Ahhh, attached are the worked sols for the Q!

I just don't understand the second part and I tried visualising it but the lines x=1 & x=4 don't intersect at all.. and so I wasn't sure why they had (4)^2 - (1)^2? (i.e. x=4 curve - x=1 curve)

[stonecold]

You need to use this formula:

V=pi x r^2 x thickness

where r=x=y^2

you are looking for the volume between x=1 and x=4, when x=1, y=1 and when x=4, y=2



One of us is wrong. :p

[Andiio]

You need to use this formula:

V=pi x r^2 x thickness

where r=x=y^2

you are looking for the volume between x=1 and x=4, when x=1, y=1 and when x=4, y=2



One of us is wrong. :p

I...don't think my brain is working....

[stonecold]

You need to use this formula:

V=pi x r^2 x thickness

where r=x=y^2

you are looking for the volume between x=1 and x=4, when x=1, y=1 and when x=4, y=2



One of us is wrong. :p

I...don't think my brain is working....

Yeah, well as always, I left out the pi haha.

Wait for the spesh heavyweights to come and confirm. 

[luken93]

You need to use this formula:

V=pi x r^2 x thickness

where r=x=y^2

you are looking for the volume between x=1 and x=4, when x=1, y=1 and when x=4, y=2



One of us is wrong. :p

Yeah, that's what I get as well...

Which question is this anyway andiio?

[Andiio]

Q6(b) in Maths Quest haha.

Urghhh does this mean the worked solutions are being retarded?

Q: For the regions bounded by the x-axis, the following curves, and the given lines, find the volume generated by rotating it about the y-axis.
y=√x; x=1, x=4

[stonecold]

Oh, yeah, it is a different question.   The solutions are correct.

You have to split it into two separate integrals, because between y=0 and y=1 you are integrating between straight lines, and between y=1 and y=2, you are integrating between a straight line and a curve.

Draw it out and you should see.

[Andiio]

Oh, yeah, it is a different question.   The solutions are correct.

You have to split it into two separate integrals, because between y=0 and y=1 you are integrating between straight lines, and between y=1 and y=2, you are integrating between a straight line and a curve.

Draw it out and you should see.

Ahhh, I get that for when integrating between y=1 and y=2, it's a straight line and a curve, but don't you integrate between a line and a curve for between y=0 and y=1 as well? Or am I thinking completely wrong? I've shaded the areas that the curve itself makes WITH the y-axis :S

[kamil9876]

See the attatched diagram, we want to find the volume generated by rotating the pink region. What Stonecold actually did was find the volume generated by the blue region.

The idea behind the solution you posted is to find the volume generated the pink part enclosed by y=0 and y=1 (that's the first integral) then the second integral finds the other pink part.

[stonecold]

Oh, yeah, it is a different question.   The solutions are correct.

You have to split it into two separate integrals, because between y=0 and y=1 you are integrating between straight lines, and between y=1 and y=2, you are integrating between a straight line and a curve.

Draw it out and you should see.

Ahhh, I get that for when integrating between y=1 and y=2, it's a straight line and a curve, but don't you integrate between a line and a curve for between y=0 and y=1 as well? Or am I thinking completely wrong? I've shaded the areas that the curve itself makes WITH the y-axis :S

Yeah, you shaded the wrong bit.  You should have the area under Y=SqRoot(x) shaded in between x=1 and x=4.  this is what you rotate.

And it is actually an anulus, so r^2=r^2 outer - r^2 inner

Edit:  The pink bit in kamil's diagram :p

[Andiio]

Ohhh... right.. I shaded in the blue part and half of the green part :| I got it confused with the ares with the y-axis then?

So basically for this question I just calculate the area bound between x=1 and x=4... wait is the green supposed to be pink in the above diagram?