Volumes quick question (1 Viewer)

[Aysce]

Hello.

Q. Rotate y=4-x^2 around the x-axis by taking slices perpendicular to the axis of rotation and find the volume enclosed.

I know the process to do it but looking at the diagram of the solution, I am unsure of how the parabola intersects the y-axis at 4 and -4, and why it intersects the x-axis at x=2.



Thanks in advance!

 

[SpiralFlex]

It doesn't, it's the rotated disk (vertex of the parabola that does.). You're looking at it sideways, you are meant to look at it as a concave down parabola.

The part that "cuts" the y intercept is the rotated figure.

Use the annulus and solve via disks ^_^

 

[Aysce]

Ermm I don't understand what you're saying? Sorry but how do you even know the values of where they "cut"?

 

[SpiralFlex]

The parabola clearly cuts at x=2, it is concave down.

You were looking as if "the parabola was sideways". This is not a parabola. This is the final figure as it is rotated about the x axis.

 

[Carrotsticks]

Sigh... I never understand why the HSC forces us to do the lim as delta x approaches zero sum of etc etc.

So much easier to write the integral...

 

[SpiralFlex]

Sigh... I never understand why the HSC forces us to do the lim as delta x approaches zero sum of etc etc.

So much easier to write the integral...

It's to get us to "appreciate" where it comes from rather than just quoting the formula.

 

[Carrotsticks]

It's to get us to "appreciate" where it comes from rather than just quoting the formula.

But the definition of the integral is EXACTLY lim as etc etc... it's the same thing. I would have thought teachers would already define the integral (or at least outlined it).

The unfortunate thing is that when integration is initially taught, it is just quoted as a formula rather than a sum of infinite infinitely small rectangles (does that make sense to you? lol) and suddenly when we go into Extension 2 (where 'appreciating it' becomes almost banal), we are forced to write it as the limit etc etc.

OP: The diagram is showing the intercepts of the rotated figure too, not just the original curve.

 

[Aysce]

OHHH okay I see now. Wow.. I kept thinking we had to turn the parabola sideways and whatnot. I don't like volumes...

 

[Carrotsticks]

OHHH okay I see now. Wow.. I kept thinking we had to turn the parabola sideways and whatnot. I don't like volumes...

The most useful tool is your ability to visualise things.

Imagine your very own portable MatLab... in your head.

 

[Aysce]

Yeah but I just get confused with some of the processes. like you take a strip perpendicular to axis of rotation, find the area of the shape the strip is taken from and integrate for volume. But some of my diagrams just turn out to be nasty and I have no clue what to do.

 

[SpiralFlex]

1. Make sure you have the right diagram. Label all important points.

2. Work out if you need an annulus.

3. Make sure your slice is PERPENDICULAR to the axis of rotation. (In this case)

4. Make sure you find delta x and all necessary variables before integrating.

5. Make sure you are integrating with respect to some variable correctly.

6. Don't forget pi and units cubed.
[HR][/HR]
With adequate exposure you will be able to do any volumes question and appreciate its intrinsic nature.

 

[mirakon]

Sigh... I never understand why the HSC forces us to do the lim as delta x approaches zero sum of etc etc.

So much easier to write the integral...

This

 

[Shadowdude]

Sigh... I never understand why the HSC forces us to do the lim as delta x approaches zero sum of etc etc.

So much easier to write the integral...

They did? I just remember I went straight to the integral.

 

[mirakon]

They did? I just remember I went straight to the integral.

I'm pretty sure you lose marks if you do that

 

[Shadowdude]

I'm pretty sure you lose marks if you do that

Damn, so maybe my 59/120 in MX2 would've been a 60/120 if I did the limit of the sum thing =P


Also I remember I went to this MX2 enrichment day and they used integrals as a sort of limit of the sum of infinitely small rectangles... and to be honest, my school didn't really use that at all.

I was just taught integration at school to be more of a mechanical process, anti-differentiation, that had the magical property of being able to find areas under curves.

 

[mirakon]

Damn, so maybe my 59/120 in MX2 would've been a 60/120 if I did the limit of the sum thing =P


Also I remember I went to this MX2 enrichment day and they used integrals as a sort of limit of the sum of infinitely small rectangles... and to be honest, my school didn't really use that. I was just taught it to be more of a mechanical process, anti-differentiation that had the magical property of being able to find areas under curves.

Our school uses cambridge in prelim which when introducing integration actually explains it via the rectangles, so it became even more stupid in ex2 to explicitly point out the limit thing for volumes

 

[Shadowdude]

Our school uses cambridge in prelim which when introducing integration actually explains it via the rectangles, so it became even more stupid in ex2 to explicitly point out the limit thing for volumes

I can't remember if Fitzpatrick (what I used) uses that same explanation. And then for 2u we used Maths in Focus or something... so I dunno.

Sounds pretty stupid though, considering I only actually fully learned the "integrals are the sum of rectangle areas" thing last semester when we were doing Riemann integrals.

 

[Aysce]

I think MIF had the sum of rectangles but it was very basic. Just like a picture and 2 lines explanation iirc? But it doesn't really matter... does it?

 

[Shadowdude]

Well for you, yes - because apparently you'll need it for your HSC.

 

[Aysce]

Like I kinda get the concept, because in volumes you take a strip out of n slices and to get the sum volume you integrate which in other words is the sum of all n slices. Pretty sure that's how it works