For questions related to volume.

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0
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1answer
50 views

Volume calculating using double integral

Here is my task: Calculate the volume under the surface $z=x^{2}-y^{2}$ over the region $(x^{2}+y^{2})^{3}=a^{2}x^{2}y^{2}$. Before solving this task, let's say that $z=x^{2}+y^{2}$ instead ...
1
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0answers
28 views

Calculating the volume bounded by $z = 5$ and $z^2=x^2+y^2$ in 2 ways

I don't understand where is my mistake on calculating the volume by the second way. The volume that I want to calculate is bounded by $z = 5$ and $z^2=x^2+y^2$, so it is the upper part of the cone, ...
0
votes
0answers
25 views

Volume bounded by $y = \sqrt{25-x^2}, y=0, x=2, x=4$ about the $x$-axis [closed]

I have come up with this integral $$2\pi \int_3^{\sqrt{21}} y\left(4-\sqrt{25-y^2} \right)\, \mathrm{d}y + 2\pi \int_0^3 2y\, \mathrm{d}y$$ ...
1
vote
1answer
17 views

Evaluating the volume of a torus formed by rotating a region about a horizontal axis using shells.

Using the method of cylindrical shells, find the volume of the shape created by revolving the region $x^2+(y-5)^2=4$ about $y=-1$. A cylindrical shell is given by: $2\pi v f(v) \ dv$ I solve ...
0
votes
1answer
15 views

Evaluate the volume of a solid of revolution using shells.

A cylindrical shell $S$ formed by some revolution about the $y$-axis is given by the equation: $S=2\pi x f(x)dx$, where the circumference $C$ of the shell is $C=2\pi x$, the height of the shell ($H$) ...
0
votes
1answer
30 views

Volume bounded by the regions $y= \frac{1}{x}, x=1, x=2, y= 0$ about $x= 3$ using the shell method

Find volume by these bounded regions $y=\dfrac{1}{x}, x=1, x=2, y= 0$ about $x= 3$ (shell method) Not sure what is wrong with my integral here. $$2\pi \int_1^2 \frac{(3-x}{x} \, \mathrm{d}x+ ...
1
vote
1answer
47 views

What is the depth of water above the prism?

I have been practising for a math competition and came across the following question: A fishtank with base $100\,\rm cm$by $200\,\rm cm$ and depth $100\,\rm cm$ contains water to a depth of ...
-1
votes
2answers
38 views

Find the volume formed by rotating the region bounded by $y = e^{-x} \sin x$, $x\ge 0$ about $y =0$.

Find the volume formed by rotating the region bounded by $y = e^{-x} \sin x$, $x\ge 0$ about $y =0$. I tried to graph this using Wolfram Alpha, but it didn't help. I don't know how to start or ...
0
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0answers
18 views

$Y = e^{-|x|}$,$ x\geq 0$ about $y = 0$. Use disk and shell method

Find the volume bounded by $y = e^{-|x|}$, $x\geq 0$, about $y= 0$. Use disk and shell method. Here is my integral attempt http://www.wolframalpha.com/input/?i=integral+0+to+1+%28y%29%28-lny%29 ...
0
votes
1answer
22 views

Find the volume of the solid obtained by rotating the region [closed]

Find the volume of the solid obtained by rotating the region in the $xy$ -plane enclosed by the parabola $y^2=x$ and the line $x=10$ around the $x$-axis.
0
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0answers
44 views

Volume between $y = 1/x$, $y= 0$, $x=1$, $x=3$ about $y = -1$ (using shell method)

Pretty obvious here that disk method is easy and I got the right answer according to the book with it. However for the last hour I have been trying to use shell method and nothing seems to be working. ...
-5
votes
0answers
47 views

what's the volume e? [closed]

I have below picture. There are two cones with an intersection in between. I want to calculate the common area "e". What other information is needed? are these enough to calculate the common area "e"? ...
0
votes
3answers
69 views

How do I derive the volume of this cup?

How do I derive the volume of this cup? It's been many years since I've taken calulus... So far I've started with the radius of the bottom and integrated that around the circle. Did I start it right? ...
0
votes
1answer
23 views

Calculating the volume of a solid of revolution about a line.

A figure is formed by revolving the region bounded by $f(x) = \cos{(x)}$ and $g(x) = \sin{(x)}$ from $0$ to $\dfrac{\pi}{4}$ about the line $y=-1$. This figure is formed by integration of two ...
9
votes
1answer
85 views

Volume of the intersection of two cylinders

I have two infinite cylinders of unit radius in $\mathbb{R}^3$, whose axes are skew lines. Say that the axis of one is centered on the $x$-axis, and the axis of the other is determined by the two ...
0
votes
1answer
17 views

Finding the volume of a cone with and oblique base.

The base of $S$ is an elliptical region with boundary curve $9x^2+4y^2=36$. Cross-sections perpendicular to the $x$-axis are isosceles right triangles with hypotenuse in the base. The base of $S$ is ...
0
votes
1answer
41 views

Calculating the volume of cutted cone by a plane

I need to calculate the volume bounded by the plane: $x+y+z=5$ and by the cone $z^2 = x^2 + y^2$, som my V that i'm $dv$-ing on it is cutted cone in non simetric way (i can find the equation of the ...
0
votes
1answer
52 views

Find the volume of an object using integrals

How am I supposed to find the volume of an object when I know that: $$x^2+y^2\le z^2, \ 0 \leq x, \ 0 \leq y, \ 0 \leq z \leq 1$$
5
votes
2answers
49 views

calculating the volume of a room with a lopsided ceiling

Part of my job description requires that I find the volume of a room for calculating air leakage. Normally no problem, but this is an unusual house for many reasons. The main issue I'm having a ...
0
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0answers
22 views

Shared Volume of Overlapping 3D Cubes/Rectangles

Good afternoon, I have been looking for an approach to figure out the volume where two cubes/rectangles overlap, meaning, I know when they do, I just don't know the coordinates of the volume in which ...
1
vote
1answer
25 views

Calculate the volume of mass in a burial mound

I am trying to calculate the volume of some burial mounds, but I am embarrassingly poor at math. The shape of an ideal burial mound is most similar to a hemisphere. I have read other people's work ...
1
vote
1answer
20 views

Minimum Surface Area of a Closed Cylindrical Container

This is a trivial question; but I just want to make sure: A closed cylindrical container has a capacity of $128\pi \,{\rm m}^3$. Determine the minimum surface area. The answer is $96\pi$. Volume of ...
0
votes
1answer
30 views

Find the volume generated by rotating the given region about the specified line

I am really unsure how to solve this problem. I was hoping someone can give me some helpful hints so I can figure it out. I also looked at some other posts but they weren't helpful Find the volume ...
1
vote
1answer
20 views

Negative volume of solid of revolution around x axis.

There was a question on my exams yesterday about the volume of revolotion of the solid generated when the circle $x^2+(y-3)^2=1$ rotates around x axis. That moment I thought it would be best if I used ...
0
votes
1answer
32 views

Points in a given volume/Area

I have a rectangular prism(3D bounding box) for which i have the point(i.e center of gravity) and the height,width,depth dimensions . Given these parameters, is it possible to find all the points that ...
0
votes
1answer
13 views

Weight on a specific point

I have a weight of 27tonnes and it has 6 points of contact with the ground. The weight is not evenly distributed, so how do I work out how much weight is on each of the 6 points? I know where the 6 ...
0
votes
1answer
17 views

Integrating Formula of spherical cap volume with respect to $h$

Context: http://tutorial.math.lamar.edu/Classes/CalcI/MoreVolume.aspx (Example 3) When integrating the spherical cap volume formula, Paul comes up with the following solution: ...
1
vote
1answer
21 views

For cylindrical shells, in which order do we take the limits?

Consider the region bounded by the $y$ axis, $y=3-x^2$, $y=x+x^2$ and $x=-1$. This region is rotated around the $y$ axis to form a solid. Now I've constructed my shell etc and acquired the integral ...
0
votes
1answer
24 views

Using differentials to find an approximate formula for percentage increase

I have some past exams that I'm practicing before the real thing. They didn't come with answers and I'm really stumped on this one question: "The volume V of a cylinder of radius r and height h is ...
2
votes
2answers
49 views

different ways to calculate the volume of a ball with radius $R$

We all know the volume is $\dfrac{4\pi R^3}{3}$. the 2 regular of ways to reach that are : (1) triple integral (there are at least 2 ways of solving the integral). (2) Solid of revolution. I have to ...
0
votes
1answer
26 views

Find a volume of a figure given by an astroid rotating around an axis

In the class we were given a task to find a volume of a figure of revolution. The figure is an astroid $x=a\cos^3{t}, y=a\sin^3{t}$ axis is $x=a$. And I thought that instead of doing integration (wich ...
3
votes
0answers
37 views

Lebesgue measure of a parallelepiped

Suppose we have $n$ linearly independent vectors $\mathbf{x}_1$, $\cdots$, $\mathbf{x}_n$ in $\mathbb{R}^n$. Let $\mathbf{X}$ be the $n \times n$ matrix with column $k$ given by $\mathbf{x}_k$, $k = ...
0
votes
1answer
47 views

Double integral with three variables

I have a difficulty calculating the volume of a solid that is between $$z=0,y=0,x=0, x+y+z=2 , y^2=1-z$$ . How can I work this with double integral cause draw this its difficult.
0
votes
0answers
16 views

Integral bounderies for solid volume

Im solving a question to evaluate a solid volume using cylindrical coordinates, the question is : Consider a solid bounded by$ x\ge0 \, y \ge0 \, z \ge0 $, the sphere $ x^2 + y^2 +z^2=1$ and the ...
3
votes
2answers
34 views

Using spherical coordinates to find volume of a region

Use spherical coordinates to find the volume of the region lying above $z = \sqrt{3x^2+3y^2}$ and within the $x^2+y^2+z^2=2az$, $a>0$. So far I know that the first graph is a cone and the second ...
1
vote
1answer
32 views

Finding volume of a cone using triple integral

The cone has the formula: $x^2 + y^2 = z^2 , 0≤z≤2$ So I used the cylindrical coordinates to get the following answer: $$\int_0^{2\pi}\int_0^2\int_0^2 dz\,rdr\,d\theta = 8\pi$$ In the solution of ...
0
votes
1answer
17 views

Calculate rotational volumes

I need to calculate the volume from rotating f(x) around y=2x using Pappus–Guldinus theorem. For that I need to know the distance A. $$L = (f(x) - 2x) / 2$$ But how can I optain the distance A?
0
votes
1answer
24 views

finding volume of an n-dimensional pyramid numerically

In my experiment I need to compute hypervolume/area from a set of points, let's start with a base case -- Triangle: In this case, I have 3 points in a 2D space and they make a triangle, $p_1 = ...
0
votes
2answers
39 views

volumes solving for dx or dy

The only problem I have with this is knowing when you are solving for dx or dy. For example, this question which says find the volume of the solid created by rotating the region bounded by y = 2x-4, ...
-2
votes
1answer
27 views

volume of a triangular pyramid [closed]

A point M0 with positive coordinates (x0, y0, z0) (x0>0, y0>0, z0>0) is given. A plane is drawn through this point. Find the minimal possible volume of a triangular pyramid whose faces lie in the ...
1
vote
0answers
33 views

Volume between a cone and a sphere

I am looking for the volume between a cone : $x^2 + y^2 = 2(95-z)^2$ and a sphere : $(x-100)^2 + (y-100)^2 + (z-150)^2= 125^2$. I want to determine the volume of their intersection. I tried to change ...
0
votes
2answers
29 views

finding volume of solid

Suppose that a solid is formed in such a way that each cross section perpendicular to the x-axis, for $0 \le x \le 1$, is a disk, a diameter of which goes from the x-axis out to the curve $y = ...
3
votes
2answers
51 views

Cylinder cut out of a sphere. (volume).

1.) A cylinder $Z=${$(x,y,z)\in R^3|x^2+y^2\leq \rho^2$}$~~$ ($0<\rho<R$) is cut out of a sphere $K\in R^3$ with radius $R>0$ centered around the origin. Find the volume of the rest ...
2
votes
1answer
28 views

Finding volume using washer method

I'm supposed to determine the volume of the region obtained by revolving the region lying below the graph of the given function and above the $x$-axis about the specified axis. The problem I'm given ...
2
votes
1answer
30 views

Calculating volume by shell integration

$y = \ln x$, region is delimited by $y = -1$, $y = 2$ and the $y$-axis, it rotates around the $y$-axis. It's quite simple to solve by using disk integration but I can't get it right with shell ...
0
votes
1answer
24 views

n-simplex volume and triangle.

For $n\in N$ let $\sum_n(1)$ be the standard-simplex. Let there be a point $b\in R^n$ and a basis {$a_1,...,a_n$} of $R^n$. The $n-simplex$ set up in this point b by the basis is the set ...
2
votes
2answers
84 views

Curve fitting the cross sectional area of a cake.

For my final Calculus project I have to find the area of a Bundt cake through the use of cross sectional areas. (Cakeulus) While most seniors in High School who run into this popular calculus project ...
0
votes
1answer
22 views

Finding volume of a revolution

I want to find the volume of the revolution that occurs when the region bounded by $y = x^2$ and $y = 1$ is revolved around the line $y=2$. The problem is that it is not solid and I cannot understand ...
0
votes
3answers
56 views

If the surface area of a box is 32 and its volume is doubled what is the new surface area? [closed]

Original surface area :32 Original volume: x New volume: 2x What is the new surface area? Please provide an explanation or show work, I don't know how to do it.
2
votes
1answer
20 views

Determine the volume of a solid given specific bounds

Determine the volume of the solid enclosed by the paraboloid $z = x^2 + y^2$ and the plane with equation $4x − 2y + z = 0$. Could someone explain to me whether I use double integral polar coordinates ...