Finding the volume bounded by surface in spherical coordinates

$R=4-1\mathrm{cos}(\varphi )$

$R=4-1\mathrm{cos}(\varphi )$

kokomocutie88r1
2022-07-22
Answered

Finding the volume bounded by surface in spherical coordinates

$R=4-1\mathrm{cos}(\varphi )$

$R=4-1\mathrm{cos}(\varphi )$

You can still ask an expert for help

Carassial3

Answered 2022-07-23
Author has **9** answers

Explanation:

$V={\int}_{0}^{2\pi}{\int}_{0}^{\pi}{\int}_{0}^{4-\mathrm{cos}\varphi}{R}^{2}\mathrm{sin}\varphi \phantom{\rule{thickmathspace}{0ex}}dRd\varphi d\theta =\frac{272\pi}{3}$

$V={\int}_{0}^{2\pi}{\int}_{0}^{\pi}{\int}_{0}^{4-\mathrm{cos}\varphi}{R}^{2}\mathrm{sin}\varphi \phantom{\rule{thickmathspace}{0ex}}dRd\varphi d\theta =\frac{272\pi}{3}$

asked 2022-07-17

Triple integrals-finding the volume of cylinder.

Find the volume of cylinder with base as the disk of unit radius in the xy plane centered at (1, 1, 0) and the top being the surface $z=((x-1{)}^{2}+(y-1{)}^{2}{)}^{3/2}.$.

I just knew that this problem uses triple integral concept but dont know how to start. I just need someone to suggest an idea to start. I will proceed then.

Find the volume of cylinder with base as the disk of unit radius in the xy plane centered at (1, 1, 0) and the top being the surface $z=((x-1{)}^{2}+(y-1{)}^{2}{)}^{3/2}.$.

I just knew that this problem uses triple integral concept but dont know how to start. I just need someone to suggest an idea to start. I will proceed then.

asked 2022-07-22

Using Differentials to Calculate the Volume of a Square Pyramid

Use differentials to solve the problem:

The Louvre Pyramid is a tourist attraction in Europe. It is a square pyramid, with a height of 21m, and base of side length 35m. The four faces of this pyramid are covered in glass, of thickness 0.03m. Find the volume of glass used to construct the exterior of the Louvre.

I know that the volume of a square pyramid is: $V=\frac{{a}^{2}h}{3}$, where a is its base length and h is its height.

I then solved for $dV=\frac{{a}^{2}\text{}dh+2ah\text{}da}{3}$, but I am stuck in this equation because there are many unknowns.

The next step I can think of is finding the volume, which can be in the form of: ${V}_{Glass}=V(Value+0.03)-V(Value)$, which can be evaluated like: $L(x)=f({x}_{0})+{f}^{\prime}({x}_{0})(x-{x}_{0})$, but I am not sure of the way of finding the V(Value).

Use differentials to solve the problem:

The Louvre Pyramid is a tourist attraction in Europe. It is a square pyramid, with a height of 21m, and base of side length 35m. The four faces of this pyramid are covered in glass, of thickness 0.03m. Find the volume of glass used to construct the exterior of the Louvre.

I know that the volume of a square pyramid is: $V=\frac{{a}^{2}h}{3}$, where a is its base length and h is its height.

I then solved for $dV=\frac{{a}^{2}\text{}dh+2ah\text{}da}{3}$, but I am stuck in this equation because there are many unknowns.

The next step I can think of is finding the volume, which can be in the form of: ${V}_{Glass}=V(Value+0.03)-V(Value)$, which can be evaluated like: $L(x)=f({x}_{0})+{f}^{\prime}({x}_{0})(x-{x}_{0})$, but I am not sure of the way of finding the V(Value).

asked 2022-07-26

What is the volume of a right right pyramid whose base is a square with a side 6m long and whose altitude is aqual to base side? wich one is A 36M OR B 72M OR C 108M OR D 216M 3 SQUARE WICH ONE IS IT

asked 2022-09-19

Finding volume of a sphere using integration

I have searched and found 2 methods of finding volume using integration:

- considering a small cylindrical element and integrating that over the radius

- considering a small circle element and using the relation ${x}^{2}+{y}^{2}={r}^{2}$ and integrating it over the z-axis.

I was trying to find the integration by considering a small circle element (with radius r) and using the relation $r=R\mathrm{cos}\theta $ where R is the radius of the sphere / hemisphere.

So I was thinking of calculating the volume of the hemisphere by integrating the $\pi {R}^{2}{\mathrm{cos}}^{2}\theta d\theta $ from $\theta $ to $\pi /2$. Is this method right? And how will the integration be like?

I have searched and found 2 methods of finding volume using integration:

- considering a small cylindrical element and integrating that over the radius

- considering a small circle element and using the relation ${x}^{2}+{y}^{2}={r}^{2}$ and integrating it over the z-axis.

I was trying to find the integration by considering a small circle element (with radius r) and using the relation $r=R\mathrm{cos}\theta $ where R is the radius of the sphere / hemisphere.

So I was thinking of calculating the volume of the hemisphere by integrating the $\pi {R}^{2}{\mathrm{cos}}^{2}\theta d\theta $ from $\theta $ to $\pi /2$. Is this method right? And how will the integration be like?

asked 2022-09-20

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.)

asked 2022-09-11

The length of a cuboid is 60 cm. Its height is 40% of its length and 3\4 of its width. calculate the volume of the parallelepiped.

asked 2022-11-12

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Using Symmetry for finding volume

I have a confusion regarding the symmetry of the volume in the following question.

Find the volume common to the sphere ${x}^{2}+{y}^{2}+{z}^{2}=16$ and cylinder ${x}^{2}+{y}^{2}=4y$.

The author used polar coordinates $x=rcos\theta $ and $y=rsin\theta $ and does something like this:

Required volume $V=4{\int}_{0}^{\pi /2}{\int}_{0}^{4sin\theta}(16-{r}^{2}{)}^{1/2}rdrd\theta $. The reason for multiplying by 4 is the symmetry of the solid w.r.t. xy-plane.

My point of confusion is that this solid cannot be cut into 4 identical parts, so how it can be multiplied by 4?

I have a confusion regarding the symmetry of the volume in the following question.

Find the volume common to the sphere ${x}^{2}+{y}^{2}+{z}^{2}=16$ and cylinder ${x}^{2}+{y}^{2}=4y$.

The author used polar coordinates $x=rcos\theta $ and $y=rsin\theta $ and does something like this:

Required volume $V=4{\int}_{0}^{\pi /2}{\int}_{0}^{4sin\theta}(16-{r}^{2}{)}^{1/2}rdrd\theta $. The reason for multiplying by 4 is the symmetry of the solid w.r.t. xy-plane.

My point of confusion is that this solid cannot be cut into 4 identical parts, so how it can be multiplied by 4?