# Questions tagged [volume]

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

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### Calculate the volume of rotation of $C: x = t^2, y = 2t$ and the line $y = x$

I am working on a problem (Pearson Core Pure Mathematics, Book 2, Chapter 4 Mixed Questions, Challenge) and cannot understand where I am going wrong. Below I will post 2 separate methods of mine, ...
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### Why is the volume of the solid of revolution between two graphs $\int_{0}^{b}\pi[f^2(x)-g^2(x)]dx$ instead of $\int_{0}^{b}\pi[f(x)-g(x)]^2dx$?

Let $f$ and $g$ be two non-negative functions which are integrable over the interval $[0,b]$ and satisfy $f \geq g$ on $[0,b]$. I am told that the volume of the solid of revolution created by rotating ...
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### Finding the volume of liquid in a spherical tank

Suppose there is a spherical tank partially filled with a liquid so that the liquid has a maximum depth of h. I want to find the equation for the volume of liquid in a tank in terms of depth h (and R ...
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### Why is the greedy algorithm philosophy to finding the maximum right cylinder inside a rectangular box not correct?

The largest solid right cylinder possible is placed in a rectangular box measuring 8 by 10 by 12 inches. If the box is then closed, approximately what percentage of the box's volume is occupied by the ...
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### Volume of region bounded

Find the volume of the region bounded by $y-x^3+1$, $y=0$, $x=0$ and $x=1$ revolved around the $x$-axis. I got $\frac{9\pi}{14}$ but the actual answer was $\frac{23\pi}{14}$. My work was to first get ...
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### How many tablets can I fill into a bottle?

I have tablets/capsules at hand and I need to buy bottles to put these in. My question is, given that I know the dimensions of a particular tablet/capsule such as diameter and height, how can I ...
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### Crow has to drop additional $320$ or $400$ volume pebbles to raise the water of the tank to drink?

A small treat is floating in a rectangular tank measuring 8 inches wide, 10 inches long, and 12 inches high. If the water is currently 5 inches high and a smart crow can reach at most 3 inches down ...
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### How to calculate the volume of this parameterized region?

I have to calculate the Volume of a region defined by $V=\bar{x}\in \mathbb{R}^{3}:x^{2}+4y^{2}\leq z\leq 2+2x+4y$ I tried to integrate in z from $x^{2}+4y^{2}$ to $2+2x+4y$ and then using polar ...
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### How to maximize volume of tube created with a piece of paper of area $1200\ \rm{cm}^2$
I am doing a math project where I have to maximize the volume of a tube created by rolling this piece of paper. The paper has a fixed area of $1200\ \rm{cm}^2$. I have been trying to use optimization ...