# Questions tagged [volume]

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

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### Find the volume of the region intercepted by the plane $x+y+z = \frac{3a}{4}$ and the coordinate axes [closed]

Find the volume of the region intercepted by the plane $x+y+z = \frac{3a}{4}$ and the coordinate axes. Try to obtain the answer without the use of triple integrals, as I do not know how they work.
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### Let $\Omega$ be the region in the first quadrant, enclosed by $y = 0$, $y = 3x$ and $y = -x^2 + 4$. Find the volume of the solid generated...

Could you help me to see if my analysis is good or wrong? Let $\Omega$ be the region in the first quadrant, enclosed by $y = 0$, $y = 3x$ and $y = -x^2 + 4$. Find the volume of the solid generated by ...
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### Finding the number of grams of gold in a solid sphere?

I am trying to solve a question from Leaving Cert. Ordinary Level Maths 2018 Paper 2. A solid sphere is made of gold. It has a volume of $0.113$cm$^3$. Each cm$^3$ of pure gold weighs $19.3$ grams. I ...
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### Why did the author need to extend $\alpha$ to a neighborhood of $Q$? ("Analysis on Manifolds" by James R. Munkres.)

I am reading "Analysis on Manifolds" by James R. Munkres. Definition. Let $k\leq n$. Let $A$ be open in $\mathbb{R}^k$, and let $\alpha:A\to\mathbb{R}^n$ be a map of class $C^r (r\geq 1)$. ...
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### Volume of Same Region Yields Different Values

This question comes from a previous question on this site. There is the integral $$\int_{-1}^{1}\int_{0}^{\sqrt{1 - x^2}}\int_{0}^{\frac{y}{2}}f(x, y, z) \, dz \, dy \, dx.$$ Our job is to reorder the ...
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### The method of calculating the volume of a parallelepiped halved along its entire length in a multi-dimensional space

Consider a right-angled triangle as shown below, where vertex $O$ is the right angle. If we consider the midpoint of the two legs and draw a line passing through it, we have proved that the area of ...
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### Find surface which generated by revolving a line in $\mathbb{R}^3$

Problem : Let $l$ be a line which passes two points : $(1,0,0), (1,1,1)$. And $S$ be a surface which generated by revolving line $l$ around $z$-axis. Find a volume enclosed by surface $S$ and two ...
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### how to integrate in the diagonal?

Imagine we have a function f(x,y). For simplicity, let's say: $$f(x, y) = \begin{cases} c & \text{if } 0 \leq x \leq 1 \text{ and } 0 \leq y \leq 1 \\ 0 & \text{otherwise} \end{cases}$$ With ...
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### Which calculus method should I use for this investigation? What does it mean choose a suitable length for the cross section?

Your task is to determine which shape maximises the volume of water that can be held by a gutter of a fixed length of material. choose a suitable fixed length for the cross section and explore ...
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### What is the relationship between radius percentage and volume percentage for a layered sphere?

I'm attempting to form interior structures of planets. If I know the volume percentage of each layer of the planet, how would I determine the outer radius of each layer? Let's assume the planet is ...
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### Calculate the volume of the tetrahedron...

The question Let $ABXY$ be a tetrahedron such that the triangles $ABX$ and $ABY$ are isosceles right angle in $A$ with legs of $2$ cm. Calculate the volume of the tetrahedron, knowing that there is a ...
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