Calculus II Riemann Sums Problem: cone and cake layers

[Aridus92]

So I just started Calculus II and I haven't done Calculus in about a year so I've forgotten quite a bit. I'm having particular issues with this problem. I figured that it probably has to do with a relationship between the Volume of a Cylinder, volume of a cone, and Riemann sums. The problem is that I can't for the life of me find the relationship between all 3 of them. I think it might be because the letters are throwing me for a loop. Any help would be greatly appreciated. Thanks in advance!

 

[Deleted member 4993]

So I just started Calculus II and I haven't done Calculus in about a year so I've forgotten quite a bit. I'm having particular issues with this problem. I figured that it probably has to do with a relationship between the Volume of a Cylinder, volume of a cone, and Riemann sums. The problem is that I can't for the life of me find the relationship between all 3 of them. Any help would be greatly appreciated. Thanks in advance!

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Let \(\displaystyle \mathcal{T}\) denote the triangle with vertices (0, 0), (R, 0), and (0, H). If we rotate \(\displaystyle \mathcal{T}\) about the y-axis, we will get a cone with radius R and height H. The volume of this cone can be approximated by a sum of the volumes of layers of cake. Each layer, in turn, is obtained by rotating a rectangle around the y-axis. A dissections of \(\displaystyle \mathcal{T}\) into such rectangles is shown.

The widths of the rectangles, which will become the radii of the layers, are denoted r₁, r₂, ..., r_n, with r₁ corresponding to the topmost layer. (Note, however, that the rectangle whose width is r_n has no height.)

a) Find a formula for r_k, for k = 1, 2, ..., n - 1, in terms of R, H, k, and n.

b) Find the formula for the volume of the k-th layer of the cake, k = 1, 2, ..., n - 1.

c) Write the total volume of the n - 1 layers using sigma notation. (You may wish to check your formula against the result of your warmup calculation.)

d) Take the limit of the total volume of the layers as n goes to infinity. Explain your result.

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What is the volume of the bottom-layer (cake)?

What is the volume of next top layer of cake?

 

[JeffM]

So I just started Calculus II and I haven't done Calculus in about a year so I've forgotten quite a bit. I'm having particular issues with this problem. I figured that it probably has to do with a relationship between the Volume of a Cylinder, volume of a cone, and Riemann sums. The problem is that I can't for the life of me find the relationship between all 3 of them. I think it might be because the letters are throwing me for a loop. Any help would be greatly appreciated. Thanks in advance!

View attachment 8487

I am not sure what is confusing you so this answer may not help. If so, just say what about the response is missing the point. No need to be shy.

You can see that the sum of the areas of the rectangles approximates the area of the triangle, right?

As as we increase the number of rectangles that sum will approach the exact area of the triangle. Old stuff, correct?

Now if we revolve the triangle around the y-axis, it will sweep out the volume of a cone. With me there?

Now revolve one of the rectangles around the y-axis. That will sweep out the volume of a cylinder rather than a cone, will it not?

Of course we can generate a cylinder for each rectangle, each with its own volume. And if we sum the volumes of the cylinders, that will approximate the volume of the cone.

We are taking a process that worked in two dimensions and extending it to three dimensions. A sum of areas of rectangles has at the limit the area of a triangle; a sum of volumes of cylinders has at the limit the volume of a cone.

 

[Aridus92]

I think the main problem is that I'm having trouble finding a formula to work with.
For example, I've thrown this together but I'm not sure if I'm right.

. . . . .\(\displaystyle \displaystyle \Large{ \sum_{k = 1}^{n - 1}\, \pi\, R_k^2\, \left(\dfrac{R}{n}\right) }\)

My reasoning is that, for each of the "rectangles" if we rotate them, they become cylinders.
The volume of a cylinder is

. . . . .\(\displaystyle \Large{ V\, =\, \pi r^2 h }\)

so I would use that as the basis of my equation (I think). I think the main thing that's stopping me is that I'm not exactly sure what to put in the place of "h" as part A is asking for a formula in respects to R, H, k, and n.

 

[Aridus92]

The main question I think I'm having is that I'm not sure how to go about creating a formula for this. The one I've come up with is this, but I'm not sure if it's right

. . . . .\(\displaystyle \displaystyle \Large{ \sum_{k=1}^{n-1}\, \pi \,R_k^2\, \Delta H }\)

My reasoning behind this is that I'm utilizing the volume of a cylinder as a part of the formula with the height being delta H.

I think that might answer part B or A of the question if it's correct.

 

[JeffM]

This was a PM and my response.

Hey JeffM, thanks for giving me the reply.

I can't respond to your comment in the thread as I just joined the forum and need a mod's approval to post. For reference, this was the original post: https://www.freemathhelp.com/forum/threads/108310-Calculus-II-Riemann-Sums-Problem

Anyway, when I got back, I looked through the comments and figured that the formula might be something like this https://puu.sh/xyifa/271fc1838b.png . I'm not sure if this is right but I figured that since the rectangle slices make a cylinder, I'd utilize the volume of a cylinder as the main equation. I think, if this is correct, this will answer either part A or B of the problem. I think the main problem that I'm having with the question is that I'm getting confused with how exactly to deal with the different letters, such as R, H, n, and k. Thanks for any additional info you can provide.

Usually the moderators are pretty quick to approve a post. I'd suggest submitting this as a public post for a number of reasons, one of which is that I am so sleepy I can hardly see straight. A quick and superficial answer is that what you are going to be summing are volumes of cylinders. If that is what you mean by the main equation then you are correct. But how do you compute those volumes? The height of each cylinder is going to be H / n, a constant. But the radius of the different cylinders will not be constant. You need to find an expression giving the radius for the different cylinders depending on n.