For questions related to volume.

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1answer
27 views

Find the volume $z \geq 3x^2+2y^2, \ \ 3x^2+2y^2+5z^2 \le 1$

Find the volume of solid defined by the following inequalities : $$z \geq 3x^2+2y^2, \ \ 3x^2+2y^2+5z^2 \le 1$$ We have an ellipse, which the semi-axis are $\sqrt{\frac{z}{2}}$ and ...
0
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1answer
24 views

Geometry - Volume of a distorted tent

How would one calculate the volume of a tent shaped object with the upper edge not parallel with the base plane of the tent? edit: The tent has a rectangular base with two poles at different heights ...
3
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1answer
29 views

Volume when rotated about the line $y=-1$

Find the volume when the region enclosed by $y=x^2$, $y=4$ is revolved around the line $y=-1$ My teacher has given the following answer: I assume she has done this through the method of shells, ...
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1answer
28 views

Volume of solid, calculus II [on hold]

Find the volume of the solid generated by revolving the region bounded by the curve $y=e^{x+1}$, the $x$-axis, the $y$-axis, and the line $x=-1$, about the line $y=e$. Please help me...it would be ...
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1answer
40 views

Derive a formula for the volume of the wedge in terms of the constants a, b, c.

Derive a formula for the volume of the wedge in terms of the constants a, b, c. Seeing a similar triangle, I see that $\frac{x}{y}=\frac{c}{b}$, $y$ being the distance from the $a$ line to the ...
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1answer
36 views

Volume bounded by two solids

Can somebody help me get started in the right direction for this question involving volume? The question is "Find the volume of the solid region inside the hemisphere $x^2 + y^2 + z^2 =6, z<0$ but ...
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0answers
17 views

Volume of a region R when revolved about the x-axis (multiple problems)

based on this earlier question I had Volume of a region R when revolved about the x-axis region bounded by y=x^2+1,y=2,x=0 is revolved about y=-1 region bounded by y=x/2, y=sqrt(2x) is revolved ...
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2answers
46 views

How to use Monte Carlo method to find volume?

So the question asks to use the Monte Carlo method to find the volume of an irregular figure defined as: ...
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1answer
30 views

Volume of a region R when revolved about the x-axis

Find the volume of the region when revolved about the $x$ axis $y= \sqrt{x-1}$, $y=2$, $y=0$, and $x=0$ Is this right? Also if you could help me with revolving this same region around $y=2$, ...
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0answers
33 views

Volume of an irregular hexahedron, no sides equal or parallel [closed]

EDIT1: Find volume of a convex hexahedron with 12 straight sides ( a,b,c,d,e,f,g,h,i,j,k,l), assuming each vertex rigidly connects neighboring sides and each face is a skew quadrilateral of minimum ...
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2answers
41 views

Find volumes using calculus?

The volume below by $z=\sqrt{x^2+y^2}$ and above by $x^2+y^2+z^2=1$. My Solution: Wrote the integral. Converted it into cylindrical coordinates. But keep getting $0$ as my answer. Can someone help ...
0
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1answer
18 views

Use differentials to estimate the error in volume of the box.

In a manufacturing process the boxes with nominal dimensions of $5$ inches by $5$ inches by $2$ inches are subject to an error of $1\%$ in each dimension.Use differentials to estimate the error in ...
0
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1answer
26 views

Volume of a cylinder cut by a plane

I've looked online but I can't seem to find a calculus proof for the volume of a cylinder cut by a plane. The question is:...
3
votes
1answer
45 views

I don't understand a proof about volume and surface of revolution

So I am investigating volume and surface of revolution, in particular the shape called Gabriel's Horn, which is $$\int_1^\infty \frac{1}{x}dx.$$ The interesting property about this shape is that it ...
0
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1answer
26 views

Finding volume by integration

Well, the question says "The area bounde by hyperbola xy=4 and line x+y=5 is revolved about x axis. Find the volume of solid thus formed. Having known that this site doesn't solve your homework ...
3
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0answers
23 views

Correct integral to compute volume of a solid

I want to compute the volume obtained by rotating the bounded region of $y=3-x^2$, $y=2x$, $x \leq 0$ around the $y$-axis. I want to use the cylindrical shell method, so my integral is: ...
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2answers
75 views

Finding the volume bounded by surface $y^2=4ax$ and the planes $x+z=a$ and $z=0$

The problem is stated below: Let $V$ be volume bounded by surface $y^2=4ax$ and the planes $x+z=a$ and $z=0$. Express $V$ as a multiple integral, whose limits should be clearly stated. Hence ...
2
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0answers
27 views

If $f$ is integrable on $S_1$ and $S_2$ ,then $f$ is integrable on $S_1$\ $S_2$ . [closed]

Can anyone give some hint how to proceed with the problem below: Let $S_1$ and $S_2$ be bounded simple sets in $\mathbb R^n$ .Let $f:S_1\cup S_2:\to\mathbb R$ be bounded function.Show that if ...
0
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1answer
20 views

question on multiple integrals, volume

compute volume of a figure bounded by $z=2-\sqrt{x^2+y^2}$ and $z=0$. I don't know how to do this, thanks for help!
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0answers
19 views

Volume of Parallelepiped

Show that the volume of the parallelepiped spanned by vectors $v_1,\ldots,v_n$ in $\mathbb R^n$ is $\det(v_1|\ldots|v_n)$. The parallelepiped is $A=\{a_1v_1+\cdots+a_nv_n:0\leq a_i<1 \}$ The ...
2
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3answers
67 views

What is the volume bound by $x^2+y^2+z^4=1$?

I saw this question on an old exam for my calculus course, and I can't get anything (Mathematica, Matlab, etc.) to plot me a graph of the function. I'm not quite sure how to setup the triple integral ...
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0answers
13 views

Integration with change of variables for a transformed cube

Define the transformation $f(x,y,z) = (u,v,w)$ given by $u = x + \frac{1}{2}y^{2}, v = y + \frac{1}{2}z^{2}$ and $w = z + \frac{1}{2}x^{2}$ being injective on the cube $S = \left\{{(x,y,z)|0 \leq x ...
0
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1answer
22 views

Is it possible/how does one find the volume under a surface and over a region bounded by a vector-valued function?

In class we learned how to find the volume under a function f(x,y) and over a rectangular region using double integrals. We then extended this to volume over triangular and circular regions. Is there ...
4
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2answers
102 views

How much extra sea water is needed?

We might call this the “Noah's Ark” calculation, but in the movie Waterworld (1995) they have the icecaps melting and take poetic license making the ~$220$ feet or so of sea water (one estimate I've ...
2
votes
1answer
41 views

How to Evaluate this Multiple integral

$$\int\!\!\!\int\!\!\!\int_{V}\frac{(b-x)\:\mathrm{d}x\:\mathrm{d}y\:\mathrm{d}z}{\left(\sqrt{(b-x)^2+y^2+z^2}\right)^3},(b>a>0)$$ $$V:x^2+y^2+z^2\leq a^2$$ Let ...
0
votes
3answers
83 views

Volume of the solid bounded by the sphere $x^2 + y^2 + z^2 = 9$ and paraboloid $8z = x^2 + y^2$

Find the volume of the solid bounded above by the sphere $x^2 + y^2 + z^2 = 9$ and below by the paraboloid $8z = x^2 + y^2$ I'm having some trouble finding the correct limits of integration in ...
0
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2answers
27 views

Innovation behind formula for surface area and volume of a sphere

When I saw some problems about innovation behind area of a circle in this site,I was wondering that about a sphere.we know volume of sphere is $\frac{4}{3}\pi*r^3$ and surface area is $4\pi*r^2$,but ...
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0answers
21 views

Find the volume of a triangle rotated about axis

I'm not sure how to do these integrals with out formulas and rather with these ranges: Let $T$ be the triangle $1 \le x \le 2$, $0 \le y \le 3x-3$ Find the volume of the solid obtained by: a) ...
0
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1answer
14 views

Distance function to define finite ratio of hypercube volume

We know that for higher and higher dimensions the volume of a hyperball inscribed in a unit volume cube approaches zero. The ball is defined by the Euclidean distance. Can you think of a mathematical ...
0
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1answer
38 views

Integrating a solid using cartesian, cylindrical and spherical coordinates

The region $W$ is the cone shown below (see image). The angle at the vertex is $π/3$, and the top is flat and at a height of $7\sqrt{3}$. Write the limits of integration for $\int_W dV$ ...
0
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1answer
21 views

Volume of snow in my city

My town is 4.7 square miles. We're getting about 6 inches of snow. Since 1 mile is 63360 inches I multiply 63360 by 4.7 to get 297792 square inches Then for volume I multiply by the depth of snow ...
1
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0answers
22 views

volume of sub blocks?

I am trying to determine the volume of a series of cubes (with nodes of known xyz) that are intersected by a solid. The solid intersecting is a cuboid. I have looked into the marching cubes algorithm ...
0
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1answer
35 views

Problem with this question on solid of revolution

Calculate the volume of a revolution solid obtained by rotation around the x-axis, the region bounded by the graph of $y=e^x$, $-1\le x \le1$ and the x-axis. Thanks in advance, and sorry about my ...
2
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2answers
39 views

Volume of revolution generated by revolving region

How can I find the volume of the solid when revolving the region bounded by $y=1-\frac{1}{2}x$, $y=0$, and $x=0$ about the line $ x=-2$? How could I set it up? Would it be $x=2-2y$ so radius $r(y) = ...
0
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2answers
40 views

limits of integration in spherical coordinates.

Consider a cone centered about the positive z axis with its vertex at origin,a $90^{\circ}$ angle at its vertex,topped by a sphere of radius $6$.Compute the volume of region bounded by sphere and ...
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0answers
11 views

Calculate ratio of volumes in mixture with given ratio of masses

I have two components of a mixture: $a$ and $b$. I know that using $3.5 \text{ kg}$ of component $a$ and $0.5 \text{ kg}$ of component $b$ will give a proper mixture, and its density will equal $1.45 ...
2
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1answer
22 views

Changing the order of integration of triple integral

We have a solid in a first octant, $x,y,z \ge 0$, which is bounded by the coordinate planes, the plane $x + y = 2$ and the parabolic cylinder $z + {y^2} = 1$. The task is to change the order of ...
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1answer
28 views

Volume of water in a cone

Let, slant height of a cone be 6cm, and radius be 3cm and the cone be uniform. Let a uniform & solid sphere of radius 1cm be put in the cone & fill the cone by water. What is the minimum ...
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0answers
25 views

Triple integral, finding the volume between two planes and a surface in 3D

So I have tried to solve this problem, but I'm running into a problem, because the top circle (intersection of the function with z=1) when you project it onto the xy plane is smaller than the circle ...
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0answers
21 views

Explanations about the volume of a regular simplex

I'm really sorry, this may sound ridiculous but I can't understand the Wikipedia explanation about the volume of regular n-dimensional simplices, here. In particular, these two sentences make no ...
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1answer
19 views

Calculus Volume of Revolution

I have to find the volume of revolution of the region bounded by $y=1-x^2$ and $y=-1+x^2$ and $x=-2$ and $x=2$ with respect to the $y$ axis. I have manipulated the equation so it is $y=1+\sqrt{1-x}$ ...
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2answers
24 views

Question on volume of swimming pool

Swimming pool is of length 20m, wide 5m and height of the swimming pool increase from the 1.6m to 4.4m. What is the volume of swimming pool? How I approached: Area of swimming pool = Area of cube + ...
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0answers
48 views

Find the volume of the region bounded by the planes $ z=8-y^2, y = 8-x^2, x=0, y=0, z=0$

I figured out the bounds for z: $z=0$ to $z=8-y^2$ The bounds for y: $y=0$ to $y=8-x^2$ The bounds for x: $x=0$ to $x=\sqrt{8}$ (Since $8-x^2 = 0$) So, the volume by using triple integral: ...
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vote
1answer
28 views

Volume of the solid cut by a plane.

I'd like to find the volume of the following solid. The solid enclosed by the paraboloid $z=4-x^2-y^2$ and the plane $x+y+z=1$. Actually original problem is the following (I made upper ...
0
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0answers
32 views

Calculate volume in ${\mathbb{R}}^{3}$ bounded by the given function, inside the region $S$.

Let $S=\{(x,y): \frac{x^2}{a^2}+\frac{y^2}{b^2}\leq 1\}$. Verify that: $$\iint\limits_S {\left(\frac{1-\frac{x^2}{a^2}-\frac{y^2}{b^2}}{1+\frac{x^2}{a^2}+\frac{y^2}{b^2}}\right ...
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1answer
95 views

Intersection Volume

Given that P is a square pyramid whose base consists of the four vertices $(0,0,0)$, $(3,0,0)$, $(3,3,0)$, and $(0,3,0)$, and whose apex is the point $(1,1,3)$. Then let Q be a square pyramid whose ...
0
votes
1answer
21 views

Determine the best cup of coffee to have faster cooling possible

Assume that a cup of coffee is a cylinder. The coffee machine at my workplace always produces the same amount of coffee, so the volume is constant. The coffee is always really hot, so I'm looking (out ...
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0answers
83 views

Triple integral-pyramid

Let the pyramid with vertices $A(0,0,0), B(0,0,1), C(0,1,0), D(1,1,0)$. I need to find the equations of the four planes bounding the pyramid then I have to set up an integral for the volume in three ...
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2answers
24 views

Finding the volume when the region between $y=x^2$ and $y=4-x^2$ is rotated about the $x$-axis.

So I want to find the volume when the region between $y=x^2$ and $y=4-x^2$ is rotated about the $x$-axis. So I start by finding the roots where they meet, so I find : $$\int_{ -\sqrt{2} ...
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2answers
141 views

Volume of pyramid intersection

Suppose that there are two square pyramids on the $xyz$-plane. Both have base coordinates of $(0,0,0)$, $(30,0,0)$, $(0,30,0)$, and $(30,30,0)$. One pyramid has its apex at $(10,10,30)$, while the ...