[Tulian]

Got stuck on one bit of this question that I'm not sure about.
http://i.imgur.com/VwLnI.jpg

For part 2 of this question, I can see where the second part of the equation comes from , ie the integrating from 0 to 1. This'll produce the extra bit on the left however, which I'm not sure how to find , of which the area is likely pi * e^2 .

[steve10]

There are two regions here.

The 1st one is R.
The 2nd one is R2, where R2 is the area between the curve C and the y-axis from y = 0 to y = 1.

The volume of revolution of R2 about the y-axis is the term on the right of the expression (the integrated bit).

The term on the left of the expression is the volume of a cylinder, centred on the y-axis, of height = 1 and radius = e.

[Tulian]

Thanks,

Oh I think I'm picturing it wrong . So is R like this , forming a shape when rotated around it, but like this ?


And what is the formula for a cylinder again, didn't think I'd need it in C4 =/ .

Then from that, subtract the R2 ?

[steve10]

No, R is already given. The red rectangle is R + R2.

[Tulian]

(Original post by steve10)
No, R is already given. The red rectangle is R + R2.

Sorry still confused

Do I find whats on the left of the region R and subtract region R from it ? And the area on the left of R , how do I know its radius is e ? It hits the x axis way before e ?

[raheem94]

(Original post by Tulian)
Thanks,

Oh I think I'm picturing it wrong . So is R like this , forming a shape when rotated around it, but like this ?


And what is the formula for a cylinder again, didn't think I'd need it in C4 =/ .

Then from that, subtract the R2 ?

The cylinder formed will have radius and height

Hence,

[Tulian]

Sorry if I sound clueless, havent seen a question like this in C4 before

I'm finding that whole red region by using the formula .

Then subtracting the integral for the R region ?

Wouldn't that just give me the white part on the left though ?

[raheem94]

(Original post by Tulian)
Sorry if I sound clueless, havent seen a question like this in C4 before

I'm finding that whole red region by using the formula .

Then subtracting the integral for the R region ?

Wouldn't that just give me the white part on the left though ?

Integration will give the volume found by rotating the yellow region around the y-axis.


Volume of cylinder will give the volume generated by rotating the green region around the y-axis.




So subtracting the volume found by integration from the volume of cylinder gives the required answer.

[Tulian]

Oh I see, thanks both . Got to go through C3 and C4 again to make sure I can apply it/ formulas properly

I just got confused because usually if you rotate it around the x-axis it included the region you done the limits for right ? So when you rotate a region around the Y-Axis , the volume formed gives everything before the shape that the limits are for ? So in essence the volume formed is actually everything before the region being rotated ?

[raheem94]

(Original post by Tulian)
Oh I see, thanks both . Got to go through C3 and C4 again to make sure I can apply it/ formulas properly

I just got confused because usually if you rotate it around the x-axis it included the region you done the limits for right ? So when you rotate a region around the Y-Axis , the volume formed gives everything before the shape that the limits are for ? So in essence the volume formed is actually everything before the region being rotated ?

When we found volume in part 'a', we were rotating the area R around the x-axis. Note that the area R was bound by the x-axis, the curve and the limits.

Now when we rotate around the y-axis, the area will be bounded by the y-axis, the limits and the curve.

So doing gives the yellow area of the below image.



Notice that this is the area bounded by the y-axis, the curve and the limits.

Does it makes sense?

[Tulian]

Makes sense , thanks alot