# Questions tagged [volume]

For questions related to volume, the amount of space that a substance or object occupies.

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### volume of a block with known are of a square inside

The task is to figure out the volume of a block ABCDEFGH . AB(EH ...) is x and BC(also FG...)is x + 23. The third side is unknown(AE, BF ...) The area of the rectangle BCEH is 4225. sketch
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### Find the volume of a specific region of a solid of revolution about the y-axis

I have a given polynomial, f(x)=c1*x^7+c2*x^6+c3*x^5+c4*x^4+c5*x^3+c6*x^2+c7*x+c8 rotated about the y axis, which results in a nice surface: What I would like to be able to do is evaluate the volume ...
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### Calculate the volume of the solid determined by $S_1$ and $S_2$

I want to calculate the volume of the solid determined by this tho surfaces: $$S_1=\{(x,y,z)\in\mathbb{R}:x^2+y^2+z^2=R^2\}$$ $$S_2=\{(x,y,z)\in\mathbb{R}:x^2+y^2=Rx\}$$ The solid is the intersection ...
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### Finding Volume of a Torus (Spherical Coordinates)

Question: Find volume of region bounded by torus where equation of torus in spherical coordinates $(r, \theta, \phi)$ is $r=2 \sin \phi$ for $\phi \in [0, \pi]$ and $\theta \in [0, 2\pi]$. Attempt: ...
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### volume of cone,cylinder, hemisphere

volume of cone,cylinder,hemisphere So i find volume of hemisphere:2/3*πr^3 volume of cylinder is πr^2*h volume of cone is 1/3*π*r^2*h now how to proceed to solve, calculate h(height),and use the ...
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### Curse of dimensionality $2^d +1$ hyperspheres inside a hypercube

Consider a $d-$dimensional hypercube $Q$ of side length $l ∈ R, i. e. |x_i − y_i | ≤ l$ for all $x, y ∈ Q$ and all $i ∈ [d]$. Note that $Q$ has $2^d$ corners. We fill $Q$ with ($L_2-)$hyperballs the ...
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### Can you see clearly _why_ $x$ and $h$ are linearly related in a triangle?

Consider a triangle with $x \in (0,6)$ and $h \in (0, 8)$: Question: Often times in questions related to volumes I have leveraged the property that $x$ and $h$ are linearly dependent to get the ...
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### Find the volume of the solid obtained by rotating

Let R be the region in the first quadrant bounded by the curves $y = f(x) = 2x+ 1$ and $y = g(x) = 2x^2 −8x + 9$ Find the volume of the solid obtained by rotating the region R about y-axis using ...
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### A new space satellite has a smooth unbroken skin made up of portions of 2 circular cylinders of equal diameters $D$ whose axes meet at right angles.

A new space satellite has a smooth unbroken skin made up of portions of 2 circular cylinders of equal diameters $D$ whose axes meet at right angles. It is proposed to ship the satellite to Cape ...
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### Volume of Solids of Revolution with Hyperbola

The area bounded above by the line $y = 3$, below by the line $y = 0$, on the left by the y-axis and on the right by an arc of the hyperbola $9x^2 - 16y^2 = 144$ is rotated around the x-axis. Find the ...
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### Triple integral and mass

An object occupies the solid region bounded by the upper nappe of the cone $$z^2=9x^2+y^2$$ and the plane $$z=9$$ Find the total mass of the object if the mass density at (x,y,z) is equal to the ...
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### Question about Polar Coordinates and Constant Values

My questions pertains to why the following limits of integration would be the following given the following scenario:\begin{equation}\int_0^a \int_0^\sqrt{a^2-x^2} \frac{dy\ dx}{(1+x^2+y^2)^\frac{3}2} ...
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### Is there a technique to identify the r limits, and function which is being integrated?

I am given the following example of which I have the solutions the already. I just wonder how the regions which are in this particular format: \begin{equation}\iint_Rf(r,\theta)r\ dr \ d\theta\end{...