For questions related to volume.

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1answer
16 views

How to work out the equation for surface to volume ratio?

I want to work out the equation for volume of half a sphere against the surface area of the circle at the widest part of the sphere. The equation of the half sphere is (2/3) * (pi * r ^3), where r is ...
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20 views

To minimize surface area of integer cuboid of ​​the known volume

There is a cuboid (a * b * c), (a, b, c ∈ N). S (Surface area of a cuboid) = 2 * (ab + bc + ca). V (Volume of a cuboid) = a * b * c = n. I need to minimize S, provided that I specified the volume ...
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1answer
29 views

Volume of a solid bounded by surfaces - is it correct?

Could you check if my calculations and reasoning are correct. And maybe suggest a nicer way of solving this problem? We are given a solid bounded by these surfaces: $y=x^2, \ y=1, \ 2x+y+z = 4, \ ...
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0answers
30 views

Could someone help find the shell height?

I am trying to solve this problem and have been going at it for 3 hours and not getting anywhere. I think I am suppose to have everything in terms of y but the x equals functions are throwing me off. ...
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1answer
23 views

Volume of revolved solid using shell method: finding height

The problem that I am working with is: Find the volume of the solid of revolution formed by rotating the region $R$ bounded by $y = 4+ x^2,\;x=0,\;y=0,\;and\;x=1$ about the line $y=10$ I have ...
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0answers
20 views

Finding the volume of a solid from revolution

Revolve $y=4+x^2$ bounded by $x=0,$ $x=1,$ and $y=0$ around $x=8$ I have started by splitting the area in two regions and using the shell method to get the part between y=4 and y=5 with: ...
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1answer
24 views

The weight of a drilled out cylinder

Cylinder is $2.71$kg or $2710$g, diameter is $5$cm and $7.5$cm high. The volume is $147.18$cm$^3$ and the density is just the mass divided by the volume, I'm not sure if I need to convert it into ...
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1answer
57 views

Volume of solid of revolution about y-axis

I need to find the volume of a solid of revolution formed by rotating the region bounded by these curves: $y=4+x^2,$ $x=0,$ $y=4+x^2,$ and $x=1$ about the y-axis Here is the graph: ...
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0answers
20 views

shell method on both axis find volume [on hold]

the region by y=4x^2 and y=7x is to be rotated about both axes and the volume generated calculated by both the washer and the shell method? the volume is the region bounded by y=4x^2 and y=7x rotated ...
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0answers
21 views

Use shell method to find volume of solid generated by revolving the given region … [on hold]

Use the shell method to find the volume of the solid generated by revolving the region bounded by the line $y=6x+7$ and the parabola $y=x^2\ldots$ ...about the following lines A) the line ...
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0answers
25 views

Disk and Washer Method to Figure out Volumes

Well, I'm having a lot of trouble setting up these problems. Here's a problem: We have the area bounded by $y = x^2/25$ and $y =1$. It is revolving about the line $y = 2$. What is the volume ...
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1answer
35 views

find the volumes of the solids generated by revolving the region in the first quadrant. [on hold]

find the volumes of the solids generated by revolving the region in the first quadrant bounded by the curve x=5y-(5y)^3 and the y-axis about the given axes. a) the x-axis b) the line y=1 please help ...
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0answers
19 views

Volume of a tetrahedron given length of edges.

I found this method to find the volume of a tetrahedron given the length of edges on Wikipedia I found this Interesting, and was looking for a formal proof, but didn't find it anywhere. Could ...
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1answer
29 views

calculus 2: Find the volume of the solid

Find the volume of the solid obtained by rotating the region bounded by the given curves about the specified line. Sketch the region, the solid, and a typical disk or washer. $$y = 1 + ...
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1answer
25 views

Multiple integral 3 dimension

Find the volume of the body $$ v:{(x,y,z) :\quad x^2+y^2\le z \le \sqrt{2-x^2-y^2}}.$$ I really don't know what to beside that i have to do triple integral of one. My main problem is to ...
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0answers
24 views

Triple integral - volume of solid described by inequalities

I have to calculate the volume of solid described by inequalities: $$(x\leqslant y)\vee (y\leqslant z) \vee (x\leqslant z)$$ in region $[0,1]^3$. What is important, here we have conjunction. It is ...
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2answers
36 views

How do i find the volume of a cone but at different heights with different radius' [closed]

I have to find the volume of a cone at different heights which means the radius would change but i dont know how to work out how to find the radius.
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2answers
171 views

Volume of Solid Revolution

region bounded by $$y=x$$ and $$y=x^2$$ a) find the volume of the solid of revolution formed by revolving R region about the line $x=2$. b) find the volume of the solid of revolution formed by ...
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0answers
22 views

Volume of the solid whose base is the region enclosed by y=x^2 and y=1 with square cross sections

I am trying to calculate the volume between $y=x^2$ and $y=1$ that has square cross-sections. It would look like a hump with curved sides and square edges. This is the integral I thought would ...
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0answers
17 views

Integrating a three variable function

Let $\Omega$ be the region in the $xy$ plane bounded by a quarter circle of radius $a$, a straight line of slope $-1$ from $(0,b)$ to $(b,0)$, and the coordinate axes. Now consider the ...
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3answers
51 views

Calculate wine volume in a horizontal barrel using a dipstick

I suck at math, but still need a way to mark a dipstick to measure the volume of wine in a barrel. This question has been asked, but the only answer is to cryptic for me to understand! My barrel has ...
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1answer
37 views

Evaluate the integral $\iiint\limits_E x^2 \,\, \mathrm{d}V$

Where E is the region bounded by the xz-plane and the hemispheres $y=\sqrt{9-x^2-z^2}$ and $y=\sqrt{16-x^2-z^2}$. This is an exercise from my professor guide. What I tried so far: These exercise ...
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1answer
37 views

Calculate the volume of the solid

Which is generated from the common part of $$ y =x^2$$ and $$y^2=8x$$ as it rotates around $0y$. Limits of the integral should be from $0$ to $4$ right? Shouldn't the integral be $$π\int_0^4(x^2 - ...
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2answers
60 views

Find the volume below $\sqrt{x}+\sqrt{y}+\sqrt{z}=1$ in the first quadrant

I understand that we have to use transformation $$x = u^2, y = v^2, z = w^2$$ but I cannot figure out the limits. I just need a rough sketch of how to approach this. Could anyone give me some ideas?
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1answer
20 views

Volume of a solid in spherical coordinates

How might we find the volume of the solid whose surface is $\rho = \sin{\phi}^{1/3}$? Of course, the obvious way to proceed is to write the triple integral $$\int_V dV$$ taking of course $dV = ...
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3answers
16 views

Disk Method Problem, where Axis of Rotation is Shifted from the Y-Axis

Could somebody please check my work here? The textbook answer is 224pi/15, but I'm getting something different. Is it a set up error? Question: Find the volume of the solid generated by revolving ...
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1answer
14 views

use area to determine the volume and cost

Two similar chocolate cakes have surface areas 144cm² and 225cm². Given that the cost of each cake is proportional to its volume. If the larger cake costs $33.75, find the cost of the smaller cake.
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1answer
56 views

Relationship between Surface Area and Volume

Question: Is there a general relationship between surface area and volume analogous to the below examples? Example 1. Consider a ball $B$ centered at the origin of a spherical coordinate system. The ...
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3answers
48 views

Find the surface area using volume

A water tank has a surface area of $8000$ cm² and a capacity of $51.2$ liters. Find the surface area of a similar water tank which has a capacity of $21.6$ liters.
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2answers
52 views

Calculating volume of spherical wedge from parallelepiped corner

I am interested in calculating the volume of the intersection of a sphere of radius 1/2 with the corner of a parallelepiped where the angles between each edge is $\pi/3$ and has unit edge length; we ...
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2answers
44 views

Proving that the volume of a pyramid is one-third that of its corresponding prism.

Is there any way to prove that for any isosceles triangle, the volume of a solid created when that triangle is projected to a point determining the height above the angle opposite the hypotenuse is ...
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3answers
159 views

Why isn't the volume of a sphere $ π^2r^3?$

Imagine two similiar circles. We rotate one on half of the other circumference. It gives a sphere with volume of $1/2(2πr) \cdot πr^2 = π^2r^3$
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2answers
29 views

Two balls fit into a circular can.

Two balls fit into a circular can (See side-view below). What is the radius of each ball if the volume of the can is 100π cm^3?
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0answers
31 views

Is it possible to create a volumetric object which has a circle, a square and an equilateral triangle as orthogonal profiles?

This question was posed to me by a friend (formulated as creating a peg to fit perfectly into holes of these shapes), and after an experiment in OpenSCAD it seems it is not possible - either one ...
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2answers
27 views

Finding the volume of a region rotated about the y-axis.

I'm having trouble trying to find the volume of the region formed by $y = x^2-7x+10$ and $y = x+3$ rotated about the y-axis. I was able to graph it, but I'm having difficulty when trying to come up ...
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1answer
52 views

What is the error in calculated volume of cylinder, given the measurements of length and radius?

Measured length of the rod is $15 \pm 0.4$ cm and the radius of the rod is $6.1 \pm 0.2$ cm. What is the error in calculated volume? (two decimal places) What i've tried is $$V=\pi r^2 l$$ ...
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2answers
24 views

Volume of a frustum

SE users I was given the problem: Find the volume of a frustum of a right circular cone with height h, lower base radius R, and top radius r. My working looks a bit too complicated and ...
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0answers
30 views

Finding the volume of a solid

How can I calculate the volume of a solid generated by the area bounded by $y=x^2-1$ $y=1-2x^2$ when rotated about the line $x=2$
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1answer
38 views

Volume of triangular pyramid given 4 vertices

I want to calculate the colume of the polyhedra with vertices $(a_1,0,0),(0,a_2,0),(0,0,a_3),(a_1,a_2,a_3)$. My solution is: Using translation and a reflection I am considering the polyhedra ...
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1answer
74 views

Ancient calculus or thorough observation

Integration. It's the simplest way on earth with which we can derive any formula like surface area or volume of symmetrical shapes and solids (square, circle, cube etc.). But what I've been hearing is ...
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1answer
46 views

Use a triple integral to find the volume of a tetrahedron

Problem: Use a triple integral to find the volume of the tetrahedron bounded by the planes $x+y+z=1$, $x=y$, $x=0$, $z=0$. I drew (what I believe to be) the solid bounded by the regions. However, I ...
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1answer
17 views

The area of the faces of a right rectangular prism are 24, 32, and 48 square centimeters. What is the volume of the prism?

Can someone show me their work, and not just the answer? I need to learn how to do this, and showing work would be greatly appreciated.
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1answer
38 views

Volume of the solid obtained by rotating half-disk around an axis

Consider the portion of the Cartesian plan delimited, in the first quadrant, from $x=0$, from $y=0$ and from the circumference of radius $= 1$ with center in the point $(0; 1)$, and determine the ...
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1answer
30 views

Determination of a volume.

Consider, in the Cartesian plane, the square Q having vertices in the points $(-1, 0), (1, 0), (0, -1)$, and $(0, 1)$. The sections of a solid with planes orthogonal to $y=0$ are squares having two ...
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1answer
40 views

Volume of the solid with given base, whose sections with the planes orthogonal to $y = 0$ are rectangles of height $4$

Please help me to solve the following problem: Determine the volume of the solid having as base the portion of cartesian plane limited by $y = 0$ and by $y = x^{3}$ in the closed interval ...
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1answer
21 views

Find total yards with LXWXH?

I have a box I need to cover all 6 sides with a single layer fiberglass fabric. The fabric comes in 50" or 28" widths. The dimensions of the box are 34" long; 15" wide; 12" High. How do I determine ...
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1answer
41 views

Finding volume of a solid

Okay so this question comes in two parts and the second part of the question doesn't make any sense to me. Let $R$ be the region in the first quadrant that lives between the curves $f(x) = x^2$ and ...
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1answer
38 views

Finding the area of a triangle from vertices? Linear Algebra

I pretty much did this problem, but I failed to get the few last blanks where they ask the area. Its confusing, they say its half the volume of matrix (u v w) in the start of the question. which means ...
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0answers
18 views

Volume of Solid of Revolution

This problem is giving me some trouble: The region bounded by the given curves is rotated about the specified axis. Find the volume V of the resulting solid by any method. x = (y − 5)^2, x = 4; ...
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1answer
42 views

Largest volume by 5 points on a sphere

I was wondering what would be the largest volume possible of the shape you get if you would put 5 points on a sphere with radius r and "wrap a paper" around those points. Don't know what it's called. ...