What is the volume in cubic inches of a box that is 25 cm by 25 cm by 25 cm (given 1 inch=2.5 cm approx.)

Valentina Holland
2022-09-20
Answered

What is the volume in cubic inches of a box that is 25 cm by 25 cm by 25 cm (given 1 inch=2.5 cm approx.)

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asked 2022-08-16

Finding the volume of a ${\mathbb{R}}^{3}$ triangle

I have a triangle ABC defined with the points $A=(2.4,-5.4,6)$, $B=(0,1.1,3.2)$, $C=(-7.6,3,0)$.

And I'm asked to find the volume of the solid of ${\mathbb{R}}^{3}$ given by the points between the plane that is made by the points of the triangle and the plaze $z=0$ (The same points of the triangle with all the $z=0$)

I think I should use integrals but I don't know from which regions to do the triple integration.

I have a triangle ABC defined with the points $A=(2.4,-5.4,6)$, $B=(0,1.1,3.2)$, $C=(-7.6,3,0)$.

And I'm asked to find the volume of the solid of ${\mathbb{R}}^{3}$ given by the points between the plane that is made by the points of the triangle and the plaze $z=0$ (The same points of the triangle with all the $z=0$)

I think I should use integrals but I don't know from which regions to do the triple integration.

asked 2022-09-16

Evaluating the Volume of a Cupola-Shaped Set by Integration

Let $f(x,y)=1-{\textstyle \frac{{x}^{2}}{4}}-{y}^{2}$ and $\mathrm{\Omega}=\{(x,y)\in {\mathbb{R}}^{2}:f(x,y)\ge 0\}.$

Compute the volume of the set $A=\{(x,y,z)\in {\mathbb{R}}^{3}:(x,y)\in \mathrm{\Omega},0\le z\le f(x,y)\}.$

My idea is to slice the set along the z-axis, obtaining a set ${E}_{z}$ - in fact, an ellipse - and computing the volume as ${\int}_{0}^{1}{\int}_{{E}_{z}}dxdydz$.

However, I am stuck finding a way to describe ${E}_{z}$. What is the best strategy to do that?

Let $f(x,y)=1-{\textstyle \frac{{x}^{2}}{4}}-{y}^{2}$ and $\mathrm{\Omega}=\{(x,y)\in {\mathbb{R}}^{2}:f(x,y)\ge 0\}.$

Compute the volume of the set $A=\{(x,y,z)\in {\mathbb{R}}^{3}:(x,y)\in \mathrm{\Omega},0\le z\le f(x,y)\}.$

My idea is to slice the set along the z-axis, obtaining a set ${E}_{z}$ - in fact, an ellipse - and computing the volume as ${\int}_{0}^{1}{\int}_{{E}_{z}}dxdydz$.

However, I am stuck finding a way to describe ${E}_{z}$. What is the best strategy to do that?

asked 2022-09-23

Finding volume of solid

Suppose that a solid is formed in such a way that each cross section perpendicular to the x-axis, for $0\le x\le 1$, is a disk, a diameter of which goes from the x-axis out to the curve $y=\sqrt{x}$.

Find the volume of the solid.

For this I use the disk formula. So $\pi {\displaystyle {\int}_{0}^{1}(\sqrt{x}{)}^{2}\phantom{\rule{thinmathspace}{0ex}}dx.}$.

When I do this, I get $\frac{\pi}{5}$. The answer is $\frac{\pi}{8}$. What am I doing wrong?

Suppose that a solid is formed in such a way that each cross section perpendicular to the x-axis, for $0\le x\le 1$, is a disk, a diameter of which goes from the x-axis out to the curve $y=\sqrt{x}$.

Find the volume of the solid.

For this I use the disk formula. So $\pi {\displaystyle {\int}_{0}^{1}(\sqrt{x}{)}^{2}\phantom{\rule{thinmathspace}{0ex}}dx.}$.

When I do this, I get $\frac{\pi}{5}$. The answer is $\frac{\pi}{8}$. What am I doing wrong?

asked 2022-08-19

Finding the volume of a solid bounded by curves.

The region bounded by the given curves is rotated about the specified axis. Find the volume V of the resulting solid by any method.

$x=(y-9{)}^{2},x=16;\text{about}y=5$

I used the washer method in terms of y and got

$V=\pi {\int}_{5}^{13}{16}^{2}-(y-9{)}^{2}dy=\frac{8192\pi}{5}\text{which is wrong}$

Also, I am having problems with another similar problem:

The region bounded by the given curves is rotated about the specified axis. Find the volume of the resulting solid by any method.

$x=1-{y}^{4},x=0,\text{about the line}x=5$

Any help on how to properly set up these integrals would be great, thank you.

The region bounded by the given curves is rotated about the specified axis. Find the volume V of the resulting solid by any method.

$x=(y-9{)}^{2},x=16;\text{about}y=5$

I used the washer method in terms of y and got

$V=\pi {\int}_{5}^{13}{16}^{2}-(y-9{)}^{2}dy=\frac{8192\pi}{5}\text{which is wrong}$

Also, I am having problems with another similar problem:

The region bounded by the given curves is rotated about the specified axis. Find the volume of the resulting solid by any method.

$x=1-{y}^{4},x=0,\text{about the line}x=5$

Any help on how to properly set up these integrals would be great, thank you.

asked 2022-08-14

Finding a volume of a region defined by $|x-y+z|+|y-z+x|+|z-x+y|=1$.

I'm having trouble approaching this problem. Could someone maybe give me a hint or a solution, it would be so helpful.

I'm having trouble approaching this problem. Could someone maybe give me a hint or a solution, it would be so helpful.

asked 2022-08-13

Finding the volume of this intersection

How to find the volume of this:

The region common to the interiors of the cylinders ${x}^{2}+{y}^{2}=1$ and ${x}^{2}+{z}^{2}=1$ and the first octant.

I tried finding the volume via double integration and the integration gets too complicated. I want to know how to solve this via triple integration. Usually there are x, y, z′s in both equations and that I can solve with triple but I dont know what to do here. I started of doing ${x}^{2}+{z}^{2}={x}^{2}+{y}^{2}$ so $z=y$ but dont know what to do with this it doesnt look like the line of intersection is $z=y$ in the diagram given.

How to find the volume of this:

The region common to the interiors of the cylinders ${x}^{2}+{y}^{2}=1$ and ${x}^{2}+{z}^{2}=1$ and the first octant.

I tried finding the volume via double integration and the integration gets too complicated. I want to know how to solve this via triple integration. Usually there are x, y, z′s in both equations and that I can solve with triple but I dont know what to do here. I started of doing ${x}^{2}+{z}^{2}={x}^{2}+{y}^{2}$ so $z=y$ but dont know what to do with this it doesnt look like the line of intersection is $z=y$ in the diagram given.

asked 2022-08-11

How can I find the volume of a solid of revolution

Find the volume of the solid obtained by rotating the region bounded by $y={x}^{2}$ and $x={y}^{2}$. Rotating about $y=1$.

I got an intercept of those functions which was (1,1). I tried to use washer method then I got

$\pi {\int}_{0}^{1}[(x{)}^{2}-(\sqrt{x}{)}^{2}]dx$ and I took integral of the functions but my volume was not right number. I think my way to solve was not right. Could you post correct way to find the volume?

Find the volume of the solid obtained by rotating the region bounded by $y={x}^{2}$ and $x={y}^{2}$. Rotating about $y=1$.

I got an intercept of those functions which was (1,1). I tried to use washer method then I got

$\pi {\int}_{0}^{1}[(x{)}^{2}-(\sqrt{x}{)}^{2}]dx$ and I took integral of the functions but my volume was not right number. I think my way to solve was not right. Could you post correct way to find the volume?