Volumes of a Solid of Revolution By Slicing and Summation

[frog1944]

I'm slightly confused about how it works with regards to the following out of my textbook;

"If we can consider a solid to be made up of lots of thin discs stacked on top of each other, and then find an expression for the cross-sectional area of each disc, then the volume of each disc is , where is the cross-sectional area at a height and is the thickness of the disc"

 (I understand all of this fine)

"The total volume of the solid can be found by adding the volumes of all the discs; sum of all sum of all "

(I understand this)



This process can be represented by

I don't understand this? How does the finite sum , become an integral of an infinite sum?

Because the summation notation the way they have written it, it indicates a sum of only times, as it increments by 1 from a to b. Doesn't it?

In another textbook I understand how they write it as;



Because, this can be seen as a Riemann Sum, so as

But I don't understand the first textbooks notation?

Thanks
(Sorry for the poorly formatted Latex, I'm not sure how to make in inline on this forum)

[RuiAce]

If you understand the second textbook's notation then I won't work through the broken LaTeX code.

This process can be represented by

I don't understand this? How does the finite sum , become an integral of an infinite sum?

If I think about it, the only difference between the first one and the second is really just that the second one can be easily represented on the Cartesian plane. The first one is just for some arbitrary thing.

Note that whilst you should understand integration theory, this is not explicitly examinable as it's considered a proof that builds up to a result you just use.

[frog1944]

Ok, but doesn't the sigma notation mean that it increments by "1" until it gets to "b". So even with the limit, doesn't that mean you're adding a finite number of really thin rectangles? Because from my understanding, the sigma notation doesn't mean we will keep summing till we get to b, but rather we will continue to add 1 to "a" until we get to b. Does that make sense?

If I leave out the limit of a sum thing, and go straight from the delta V to the integral is that fine?

Thanks

[RuiAce]

Ok, but doesn't the sigma notation mean that it increments by "1" until it gets to "b". So even with the limit, doesn't that mean you're adding a finite number of really thin rectangles? Because from my understanding, the sigma notation doesn't mean we will keep summing till we get to b, but rather we will continue to add 1 to "a" until we get to b. Does that make sense?

If I leave out the limit of a sum thing, and go straight from the delta V to the integral is that fine?

Thanks

Like I said, it's an abuse of notation

Also yes

[frog1944]

Ok, thanks RuiAce

[frog1944]

I heard that the markers might expect you to write this line in your working out. Is this true? Would you lose marks if you didn't have it?

[RuiAce]

Personally, I always wrote it and our teacher encouraged us to. I used to think it was necessary but now I'm not so sure...

[claudiarosaliaa]

Can someone help me! I'm struggling at finding the radius in volume questions. Whats the best way to identify it?

[kiwiberry]

Can someone help me! I'm struggling at finding the radius in volume questions. Whats the best way to identify it?

Could you provide an example? I find that drawing a diagram on the graph helps to visualise it

[RuiAce]

Can someone help me! I'm struggling at finding the radius in volume questions. Whats the best way to identify it?

In all honesty, the best way to find the radius is just to draw in the shell and then compare the relevant x coordinates. If this is confusing you will definitely need to provide an example.

Alternatively, if it's slices, you should just identify the appropriate rule