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Finding the volume of an elliptical cone with double integrals
For my class I need to find the volume of an elliptical cone bounded by $z=\sqrt{9x^2+y^2}$ and the plane $z=2$. My thought process was to integrate the equation for $z=\sqrt{9x^2+y^2}$ over the region bounded by the projection onto the xy-plane. This is the integral I set up:
${\int <!--\; \int \; -->}_{-<!--\; -\; -->2}^2{\int <!--\; \int \; -->}_{-<!--\; -\; -->\frac{1}{3}\sqrt{4-<!--\; -\; -->y^2}}^{\frac{1}{3}\sqrt{4-<!--\; -\; -->y^2}}\sqrt{9x^2+y^2}dxdy$
To check my answer, I looked up and found that the volume of an elliptical cone can be found using the equation:
$V=\frac{1}{3}\mathrm{\pi <!--\; \pi \; -->}abh$
When I checked my answer I got from the double integral, I found that it is 4 times what it should be. Can anyone explain to me what I did wrong?

Finding the volume of a region below a plane and above a paraboloid (Triple integrals)
I have to find the volume below the plane $z=3-<!--\; -\; -->2y$ and above the paraboloid $z=x^2+y^2$.
Integrating by z first, it looks like the "arrow" I draw parallel to z-axis enters the region at $z=x^2+y^2$ and exits the region at $z=3-<!--\; -\; -->2y$. So ${\int <!--\; \int \; -->}_{x^2+y^2}^{3-<!--\; -\; -->2y}1dz$.
How I am supposed to find the other two integrals?

Finding the volume of revolution using the method of shells
I'm trying to find the volume of the solid generated by revolving the region bounded by $y=x^2$ and $y=6x+7$ about x-axis using the shell method. I applied the method and I got 15864/5 multiplied by $\mathrm{\pi <!--\; \pi \; -->}$ but it's not correct.
Details: I integrated ${\int <!--\; \int \; -->}_1^{49}y(\sqrt{y}-<!--\; -\; -->\frac{y-<!--\; -\; -->7}{6})dy$

The radius r of a sphere is increasing at a rate of 2 inches per minute. Find the rate of change of th volume when r = 24 inches.

Two cylinder cans have the same volume. One can is triple the height of the other. If the narrow can has a radius of 12 inches, what is the radius of the wider can?

Volume: The Disk Method AP Calculus AB
Finding the Volume of a Solid - Find the volume of the solid generated by revolving the region bounded by the graphs of the equations about the x-axis
$y=e^{-<!--\; -\; -->x},y=0,x=0,y=1$
Why would the upper limit of integration be 1?

Finding volume using triple integral when i have given set.
Find the volume of:
$V=[(x,y,z):0⩽<!--\; ⩽\; -->z⩽<!--\; ⩽\; -->4-<!--\; -\; -->\sqrt{x^2+y^2},2x⩽<!--\; ⩽\; -->x^2+y^2⩽<!--\; ⩽\; -->4x]$
I should somehow construct triple integral here in order to solve this, which means that i have to find limits of integration for three variables, but i am just not quite sure how, i assume that i should first integrate for z since we could say that limits for z are already given, but what i am supposed to do with other two variables. When i find limits, what function i am going to integrate, is it going to be just $\iiint <!--\; \iiint \; -->dzdydx$?

Volume of a Hyperboloid using triple integration/shadow method
Could someone help me find the volume of the hyperboloid
$\frac{x^2}{1817}+\frac{y^2}{1817}-<!--\; -\; -->\frac{z^2}{10914}=1$
with the limits in the $z-<!--\; -\; -->axis$ of -130 to 43? I tried using triple integration, but I'm not sure how to convert the variable the in limit of the innermost integral into a number. I would also be open to any other methods of finding the volume.

Finding the volume of $\mathcal{O}$
Let K be a number field, $r_1$ denotes the number of real embeddings and $2r_2$ denotes so that $r_1+2r_2=n=[K:\mathbb{Q}]$. Define the ring homomorphism, canonical imbedding of K, $\mathrm{\sigma <!--\; \sigma \; -->}:K\to <!--\; \to \; -->{\mathbb{R}}^{r_1}\times <!--\; \times \; -->{\mathbb{C}}^{r_2}$ as
$\mathrm{\sigma <!--\; \sigma \; -->}(x)=({\mathrm{\sigma <!--\; \sigma \; -->}}_1(x),\dots <!--\; \dots \; -->,{\mathrm{\sigma <!--\; \sigma \; -->}}_{r_1}(x),\dots <!--\; \dots \; -->,{\mathrm{\sigma <!--\; \sigma \; -->}}_{r_1+r_2}(x))$
We only consider the 'first' $r_2$ complex embeddings since the rest are conjugate of the previous ones.
So, there is this lemma that I do not know how to start, what to think:
Lemma.If M is a free $\mathbb{Z}$-module of K of rank n, and if $(x_i{)}_{1\le <!--\; \le \; -->i\le <!--\; \le \; -->n}$ is a $\mathbb{Z}-<!--\; -\; -->$-base of M, then $\mathrm{\sigma <!--\; \sigma \; -->}(M)$ is a lattice in ${\mathbb{R}}^n$, whose volume is $v(\mathrm{\sigma <!--\; \sigma \; -->}(M))=2^{-<!--\; -\; -->r_2}|\underset{1\le <!--\; \le \; -->i,j\le <!--\; \le \; -->n}{det}({\mathrm{\sigma <!--\; \sigma \; -->}}_i(x_j))|$.
The determinant on the right hand side is $\sqrt{{\mathrm{\Delta <!--\; \Delta \; -->}}_K}$ as far as I know. However, I appreciate any help to understand the proof of this lemma.
Edit: The lemma does not say about anything about $\mathcal{O}$, but it is a free $\mathbb{Z}$-module of rank n. So the lemma is more general, $\mathcal{O}$ is a specific case but I wrote the title as this way. You can either take a $\mathbb{Z}$-base for $\mathcal{O}$ and continue or work on any free $\mathbb{Z}$-module of rank n.

A childs sandbox needs to be filled with sand. The dimensions of the box are 4 ft by 4 ft and is 16 inches deep. Find the volume rounded to the nearest cubic ft.

Finding the volume of an n-dimensional simplex (recursion)
I want to find the volume of an n-dimensional simplex, i.e. determine $\begin{array}{r}{\mathrm{\sigma <!--\; \sigma \; -->}}_n={\int <!--\; \int \; -->}_{A_n}^{}\mathrm{d}\mathrm{\mu <!--\; \mu \; -->},A_n=\{(x_1,\dots <!--\; \dots \; -->,x_n)\in <!--\; \in \; -->{\mathbb{R}}^n\mid <!--\; \mid \; -->\mathrm{\forall <!--\; \forall \; -->}i:<!--\; :\; -->x_i⩾<!--\; ⩾\; -->0,x_1+\dots <!--\; \dots \; -->+x_n⩽<!--\; ⩽\; -->1\}.\end{array}$
I came up with the following informal computation: We consider
$\begin{array}{r}{A}^{\prime }=[0,1],A_{x_1}=\{(x_2,\dots <!--\; \dots \; -->,x_n)\mid <!--\; \mid \; -->x_2+\dots <!--\; \dots \; -->+x_n⩽<!--\; ⩽\; -->1-<!--\; -\; -->x_1\}\end{array}$
and conclude
$\begin{array}{rl}{\mathrm{\sigma <!--\; \sigma \; -->}}_n& ={\int <!--\; \int \; -->}_0^1({\int <!--\; \int \; -->}_{A_{x_1}}^{}\mathrm{d}\mathrm{\mu <!--\; \mu \; -->}(x_2,\dots <!--\; \dots \; -->,x_n))\mathrm{d}{x}_1\\ & \stackrel{(\ast <!--\; \ast \; -->)}{=}{\int <!--\; \int \; -->}_0^1{\mathrm{\sigma <!--\; \sigma \; -->}}_{n-<!--\; -\; -->1}(1-<!--\; -\; -->x_1{)}^{n-<!--\; -\; -->1}\mathrm{d}{x}_1\\ & ={\mathrm{\sigma <!--\; \sigma \; -->}}_{n-<!--\; -\; -->1}{[-<!--\; -\; -->\frac{1}{n}(1-<!--\; -\; -->x_1{)}^n]}_0^1=\frac{{\mathrm{\sigma <!--\; \sigma \; -->}}_{n-<!--\; -\; -->1}}{n}.\end{array}$
Solving this recursion we find
$\begin{array}{r}{\mathrm{\sigma <!--\; \sigma \; -->}}_1={\int <!--\; \int \; -->}_0^1\mathrm{d}{x}_1=1⟹<!--\; ⟹\; -->{\mathrm{\sigma <!--\; \sigma \; -->}}_n=\frac{{\mathrm{\sigma <!--\; \sigma \; -->}}_1}{n!}=\frac{1}{n!}.\end{array}$
How I can formalise the (∗) step?

Write S (surface area) of a cube interms of its volume, V. so far I have that s= 6a(for area) squared. and $v=a^3$

A prismatoid has a square and an equilateral triangular bases whose edges measure 3 cm each. If the midsection is 5 cm2 and the altitude of the solid is 6 cm, find the volume of the solid.

This sort of question I'm more checking if I'm doing itcorrectly, its been awhile since I've done something like this,always need a refresher anyways the question(s) is...
E. Coli cells are about $2\mathrm{\mu <!--\; \mu \; -->}$ m long and $0.8\mathrm{\mu <!--\; \mu \; -->}$ m indiameter:
How many E. Coli cells laid end to end would fit across thediameter of the pinhead, assuming the pinhead diameter of 0.5 mm,and what would be the volume of an E.Coli cell assuming that its acylinder.
Ok, what I've done was simple I guess, the first part whichI've assumed that "long" ment length, and I've taken the longestpart of E. Coli and did the idea of $0.5\times <!--\; \times \; -->{10}^{-<!--\; -\; -->3}m/2.0\times <!--\; \times \; -->{10}^{-<!--\; -\; -->6}m=250$ Cellsgoing across.
[Wasn't sure if it was suppose to use the othernumber, just assuming it was talking about long ways]
For the second part of the question, I just used the formulaof $V=\mathrm{\pi <!--\; \pi \; -->}{r}^{2}h$ in which I came up with
$V=(3.14)(0.4\times <!--\; \times \; -->{10}^{-<!--\; -\; -->6}m)2(2.04\times <!--\; \times \; -->{10}^{-<!--\; -\; -->6}m)=1.00\times <!--\; \times \; -->{10}^{-<!--\; -\; -->18}m$
Which something doesn't seem correct, or atleast seems oddto me.
Also, I don't know if I'm pushing the amount of questions Ican ask in a single problem, but the next part its asking for the Surface area of an E. Coli cell, and what is the surface to volumeratio of an E. Coli cell?
Which I have no idea, what type of surface area I should usefor a cell, then do I just compare the volume to the surface area,that I got in the previous problem?

A local pool used by lap swimmers has dimensions 25 yd by 20 yd and is 5 feet deep. Find the cost for filling the pool if the city charges $1.50 per 1000 gallons. Use the conversion 1 gallon = 0.134 cubic ft.

What is the lateral area of a cylinder which has an element of 8 inches and a right section with a perimeter of 24 inches?

Modeling Costs for the General Store

1. Remember the form of a quadratic equation: ax2+bx+c

2. You will use: W=-.01x2+100x+c where (-.01x2+100x) represents the store's variable costs and c is the store's fixed costs.

3. So, W is the stores total monthly costs based on the number of items sold, x.

4. Think about what the variable and fixed costs might be for your fictitious storefront business - and be creative. Start by choosing a fixed cost, c, between $5,000 and $10,000, according to the following class chart:

5. My last name is Haggins so my fixed cost would be be between $5800-$6400

If your last name starts with the letter Choose a fixed cost between
A-E $5000-$5700
F-I $5800-$6400
J-L $6500-$7100
M-O $7200-$7800
P-R $7800-$8500
S-T $8600-$9200
U-Z $9300-$10,000

6. Post your chosen c value in your subject line, so your classmates can easily scan the discussion thread and try to avoid duplicating your c value. (Different c values make for more discussion.)

7. Your monthly cost is then, W = -.01x^2+100x+c.

8. Substitute the c value chosen in the previous step to complete your unique equation pblackicting your monthly costs.

9. Next, choose two values of x (number of items sold) between 100 and 200. Again, try to choose different values from classmates.

10. Plug these values into your model for W and evaluate the monthly business costs given that sales volume.

11. Discuss results and alternate business decisions.

What is the volume of a right right pyramid whose base is a square with a side 6m long and whose altitude is aqual to base side? wich one is A 36M OR B 72M OR C 108M OR D 216M 3 SQUARE WICH ONE IS IT

Let Vr denote the sum of the first r terms of an arithmetic progression (A.P.) whose first term is r and the common difference is (2r ? 1). Let r = TVr+1 ? Vr ? 2 and Qr = Tr+1 ? Tr for r = 1, 2,

a cylinder whose height is 8 inches has a volume of 128 pi cm3. if the radius is doubled and its height is cut in half, the volume of the resulting cylinder is...
please show how to do it i do not need the answer i know the answer is 256pi cm^3