Help VOlumes (1 Viewer)

[girrawhat]

Hey guys, I have my 4u trial coming soon. It will have all topics except harder 3u.

Now the reason I have posted this thread is that I am having a lot of trouble with volumes. Is there any tips to answer these types of questions. No matter how much I try, it is killing me.

Could anyone pls explain how to do one of the questions that my teacher gave for practice.

The region bounded by the curve y=(x-1)^2 and the x- and y-axes is rotated about the line y=-0.5. Find the volume of the solid.

 

[SpiralFlex]

Method 1: Annulus discs

First of all, always draw a pretty diagram. I drew one here.



Now I have taken a strip of the area bounded between the graph and x,y axes. Our aim is to rotate this portion around the line y=-0.5. Note: The strip I have taken is always perpendicular to the line of rotation.


Notice when we rotate the figure, we end up with an approximation of a disk. In this case the disc has a hole in it.












If we make delta x very small, this will "better" our approximation of a disc. If we sum up the discs from x=0 to x=1, we would get our solid.










Note: The diagram should say -0.5 instead of 0.5 with the line y=-0.5

 

[Johnstan]

thats from a catholic paper right?
question looks really familiar

 

[SpiralFlex]

Sorry my internet shut down and half the working I did for the shell method has been deleted. I will repost it up later tonight.

 

[girrawhat]

Thankyou so much Spiralflex. It actually makes sense now. Must have been some effort to draw those diagrams. Much appreciated.

 

[girrawhat]

thats from a catholic paper right?
question looks really familiar

I do not know where my teacher pulls questions from but very well could be from a catholic paper.

 

[SpiralFlex]

No problemo just give a shout if you need any more help.

 

[girrawhat]

Man, I am badly struggling with this topic. I have another question which I think requires us to use a similar annulus method.

The area bounded by y=x^2+2 and y=2x+5 is rotated about the x-axis. Find the volume of the solid so formed.

I think the problem that I am facing is identifying what values to actually use for the radius and also if we use dy or dx.

 

[Johnstan]

well for dy or dx, it depends on where the area is being rotated about...and also what type of method is required...
for eg, using cylindrical shells requires you to take slices PARALLEL to the axis of rotation.. where as slicing/crossections require you to take them PERPENDICULAR.

for eg. The area bounded by y=x^2+2 and y=2x+5 is rotated about the x-axis.By using the method of Cylindrical shells, Find the volume of the solid so formed.
kay so its rotated about x axis, cylindrical shell is parallel to axis of rot... so we gonna do delta x. also remember that ur limits will be dependant on delta x or y.
so if u take the thickness of strip as delta x, limits need to be from x axis etc...

 

[asianese]

Just a tip (more a preference thing). Rotations about lines are mostly (sometimes, often whatever) easier done with cylindrical shells because you don't need to worry about subtracting anything.

 

[Carrotsticks]

Hahaha, I believe it was my grade that started 'Girra what???'

 

[girrawhat]

for eg. The area bounded by y=x^2+2 and y=2x+5 is rotated about the x-axis.By using the method of Cylindrical shells, Find the volume of the solid so formed.
kay so its rotated about x axis, cylindrical shell is parallel to axis of rot... so we gonna do delta x. also remember that ur limits will be dependant on delta x or y.
so if u take the thickness of strip as delta x, limits need to be from x axis etc...

So u would not recommend using annulus method? Thanks, I should be fine with identifying if dy or dx now. Just a matter of knowing how to do the same question using both methods. Could u recommend situations where it is better to use slice method over shells and vice verca?>

 

[girrawhat]

Just a tip (more a preference thing). Rotations about lines are mostly (sometimes, often whatever) easier done with cylindrical shells because you don't need to worry about subtracting anything.

would u know when it would be better to use slice method over shells? Is it only when there is no rotation?

 

[girrawhat]

Hahaha, I believe it was my grade that started 'Girra what???'

U went to girra? What year did u graduate?

 

[girrawhat]

For the question of finding the volume when area between y=4-x^2 and x-axis being rotated about the x-axis, i think i am getting destroyed by the integration. I think I did the diagram and the working right and I got it to V=4pi (integral sign upper limit 4, lower 0) y(y-4)^1/2 dy. Could someone pls help me with a quick way of solving this integration problem.

 

[asianese]

Make the substitution u=y-4.

 

[asianese]

would u know when it would be better to use slice method over shells? Is it only when there is no rotation?

slices is rotation...but i would only use it when its the area under the curve rotated about coordinate axes...shells is more powerful imho

 

[girrawhat]

slices is rotation...but i would only use it when its the area under the curve rotated about coordinate axes...shells is more powerful imho

I see, so shells is your go to method. I'm slightly stronger with shells but i'm so hesitant on which method to use as soon as I see a problem