Recent questions in Finding volume

Finding volume
Answered

Nina Bean
2022-08-13

I need to find the volume of an object set with the following function:

$x+y+z=1$

And all three axis.

So I converted the function into $z=-x-y+1$, and it gave me kind of a clepsydra, crossing x,y plane in $x=y=1$.

So the region seems to be the circle ${x}^{2}+{y}^{2}=1$.

However, changing x to r, and y to $\varphi $ gives me wrong result $\pi $, and it should be 1/6.

Is the region I've came up with correct?

Finding volume
Answered

heelallev5
2022-08-13

Given $D=[(x;y)\in {\mathbb{R}}^{2}:1\le y\le a{x}^{2}+1,0\le x\le 2/a],0<a$ let W be the region obtained by rotating D around Y axis.

A) Find the volume of W

B) Find, if possible, the a values $\u03f5(0;+\mathrm{\infty})$ so that the volume of W is a minimum, and a maximum

C) Find, if possible, the a values $\u03f5(1/3;3)$ so that the volume of W is a minimum, and a maximum

Well, what I've done so far is finding the inner and outer radius to calculate the volume in terms of Y. The inner radius would be be $\sqrt{\frac{y-1}{a}}$ and the outer would be $\frac{4+a}{a}$, which is a evaluated in the parabola. Then, the volume would be ${\int}_{1}^{\frac{4+a}{a}}(\sqrt{\frac{y-1}{a}}{)}^{2}-(\frac{4+a}{a}{)}^{2}\phantom{\rule{thinmathspace}{0ex}}dx$.

And that's the function that I have to differentiate to find its maxima and minima, which, after differentiating, is 1/a. Is ok what am I doing? How can I go on?

Finding volume
Answered

Cheyanne Jefferson
2022-08-13

You have a 6-inch diameter circle of paper which you want to form into a drinking cup by removing a pie-shaped wedge with central angle theta and forming the remaining paper into a cone. - You are given that 3 is the slant height

a) Find the height and top radius of the cone so that the volume of the cup is as large as possible.

b) What is the angle theta of the arc of the paper that is cut out to create the cone of maximum volume?

I know how to do related rates with volume, but I can't seem to figure it out with angles being cut out from a circle.

Finding volume
Answered

Jenny Stafford
2022-08-12

Find the volume of the tetrahedron with the vertices P(1,1,1), Q(1,2,3), R(3,1,2), and S(2,3,1).

Finding volume
Answered

Elisabeth Wiley
2022-08-12

It says use the method of cylindrical shells to find the volume generated by rotating the region bounded by the given curves about the specified axis $y={x}^{3}y=8$ about the axis $x=3$. I drew the graph, reflected it about the $x=3$ line, and drew a cylinder. `I figured that the radius is just $r=3-x$ and the height would just be $h={x}^{3}-0$ (since the lowest y value is zero), and plugged these into the integral of $2(\pi )(r)(h)$ from 0 to 2. However, I got the wrong answer (correct answer should be $264\pi /5$).

I have a feeling that my height may be wrong but I'm not sure why.

Finding volume
Answered

traucaderx7
2022-08-12

Let R be the area laying beneath the curve $f(x)=8-{x}^{2}$ and above the line $y=7$. Find the volume of the solid of revolution which is created when R is revolved around the x-axis.

I graphed the functions and found out I have to integrate from -1 to 1. I want to use the disc method, but I don't know how to only get the discs with radii higher than $y=7$ and lower than f(x). I thought it would be logical if the radii of the discs would be $8-{x}^{2}-7=1-{x}^{2}$, but this approach gives me the wrong answer when i plug it into the formula.

Finding volume
Answered

Passafaromx
2022-08-12

I'm having problems finding the triple integrals of equations. I guess it has to do with the geometry. Can someone solve the two questions below elaborately such that I can comprehend this triple integral thing once and for all:

Compute the volume of the solid enclosed by

1. $\frac{x}{a}+\frac{y}{b}+\frac{z}{c}=1,x=0,y=o,z=0$

2. ${x}^{2}+{y}^{2}-2ax=0,z=0,{x}^{2}+{y}^{2}={z}^{2}$

Finding volume
Answered

Matonya
2022-08-12

Suppose I want to find the volume between $z=2{x}^{2}+3{y}^{2}$ and $z=4$. Is there a way finding that with a double integral? I tried to use $4-2{x}^{2}-3{y}^{2}$ inside the integral and then convert it to polar coordinates...

Finding volume
Answered

daniellex0x0xto
2022-08-11

I'm asked to find the volume of a container of height 13.5 The container is made by rotating $y=1.5{x}^{2}$ where $0\le x\le 3$ about the axis $x=-0.5$ (the bottom is flat).

I tried finding the right volume a number of ways, but I believe my most solid attempt so far is this:

I invert the function to get $y=\sqrt{\frac{2}{3}x}$.

From this I tried to calculate $\pi \ast {\int}_{0}^{3}\sqrt{\frac{2}{3}x}dx$ which gave me $3\pi $ which I thought was correct, but it wasnt. Plotting this, I realized the height becomes 3 when I integrate the inverted function from 0 to 3, so I thought I should maybe integrate from 0 to 13.5. In that case I get $\frac{243\pi}{4}$ which is also incorrect. I'm sure I've misinterpreted what it means to rotate the function about the axis $x=-0.5$

Finding volume
Answered

vroos5p
2022-08-11

I approached this problem by trying to find the volume bounded by the paraboloid and the cylinder and then subtracting it from the volume bounded by the cone and the cylinder. But I am getting the wrong answer. I converted the all the bounds into cylindrical co-ordinates.

For finding the volume bounded by the cone and the cylinder,

Bounds of integration: $\text{}z=0\text{}$ to $\text{}z=r\text{}\text{},\text{}\text{}r=0\text{}$ to $\text{}r=2\mathrm{cos}(\theta )\text{}\text{},\text{}\text{}\theta =0\text{}$ to $\text{}\theta =2\pi \text{}$.

For finding the volume bounded by the paraboloid and the cylinder,

Bounds of integration: $\text{}z=0\text{}$ to $\text{}z=\frac{{r}^{2}}{4}\text{}\text{},\text{}\text{}r=0\text{}$ to $\text{}r=1\text{}\text{},\text{}\text{}\theta =0\text{}$ to $\text{}\theta =2\pi \text{}$.

Finding volume
Answered

dredyue
2022-08-11

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?

Finding volume
Answered

vrteclh
2022-08-11

I am trying to find the volume integral of $\rho ={\rho}_{0}\left(\frac{{R}^{2}-{r}^{2}}{{R}^{2}}\right)$ inside an ellipsoid given by $\frac{{x}^{2}}{(3R{)}^{2}}+\frac{{y}^{2}}{(4R{)}^{2}}+\frac{{z}^{2}}{(5R{)}^{2}}=1$

I've tried using jacobian to move from an ellipsoid to an unit ball by these mapping relationships $x=3Ru$, $y=4Rv$, $z=5Rw$.

But the resulting integral is still heavy

$\int {\rho}_{0}(1-(9{u}^{2}+16{w}^{2}+25{w}^{2}))60dudvdw$

Does anyone have any insight to a more elegant way.

Finding volume
Answered

rivasguss9
2022-08-11

I'm trying to find the volume of a given shape:

$V=\{\begin{array}{l}\sqrt{x}+\sqrt{y}+\sqrt{z}\le 1\\ x\ge 0,\text{}y\ge 0,\text{}z\ge 0\end{array}$

using double integral. Unfortunately I don't know how to start, namely:

$z=(1-\sqrt{y}-\sqrt{x}{)}^{2}$

and now what should I do? Wolfram can't even plot this function, I'm unable to imagine how it looks like...

Would it be simpler with a triple integral?

Finding volume
Answered

musicbachv7
2022-08-10

I´m trying to find the Volume of a Circular right cone. The height is 3 units (y-axis) and the radius is 2 units (x-axis).

So, if i Want to find the volume I have to write a equation above in terms of "y", for this situation. Right? The question ask me the equation that generates the solid. I have found $x=(2/3)y$. Am I right?

Finding volume
Answered

Bobby Mitchell
2022-08-10

The region bounded by $y=\frac{x}{\sqrt{{x}^{3}+8}}$, the x-axis, and the line $x=2$ is revolved about the y-axis. Find the volume of the solid generated this way.

Finding volume
Answered

makeupwn
2022-08-10

Set up and evaluate a triple integral to find the volume of the region bounded by the paraboloid $z=1-\frac{{x}^{2}}{9}-\frac{{y}^{2}}{100}$ and the xy-plane.

I understand I'm finding the volume of a paraboloid that forms a "dome" over the xy-plane. Moreover, I can see the paraboloid intersects with the xy-plane to form an ellipse given by $\frac{{x}^{2}}{9}-\frac{{y}^{2}}{100}=1$.

I have tried setting this up using rectangular coordinates but the integral started looking extremely messy. I then tried spherical coordinates but had trouble find the upper bound of $\rho $. Specifically, I can't seem to successfully translate the rectangular equation $z=1-\frac{{x}^{2}}{9}-\frac{{y}^{2}}{100}$ to a spherical equation and isolate $\rho $.

Finding volume
Answered

Ledexadvanips
2022-08-10

For my class I need to find the volume of an elliptical cone bounded by $z=\sqrt{9{x}^{2}+{y}^{2}}$ and the plane $z=2$. My thought process was to integrate the equation for $z=\sqrt{9{x}^{2}+{y}^{2}}$ over the region bounded by the projection onto the xy-plane. This is the integral I set up:

${\int}_{-2}^{2}{\int}_{-\frac{1}{3}\sqrt{4-{y}^{2}}}^{\frac{1}{3}\sqrt{4-{y}^{2}}}\sqrt{9{x}^{2}+{y}^{2}}\phantom{\rule{thinmathspace}{0ex}}dx\phantom{\rule{thinmathspace}{0ex}}dy$

To check my answer, I looked up and found that the volume of an elliptical cone can be found using the equation:

$V=\frac{1}{3}\pi abh$

When I checked my answer I got from the double integral, I found that it is 4 times what it should be. Can anyone explain to me what I did wrong?

Finding volume
Answered

imire37
2022-08-09

I have to find the volume below the plane $z=3-2y$ and above the paraboloid $z={x}^{2}+{y}^{2}$.

Integrating by z first, it looks like the "arrow" I draw parallel to z-axis enters the region at $z={x}^{2}+{y}^{2}$ and exits the region at $z=3-2y$. So ${\int}_{{x}^{2}+{y}^{2}}^{3-2y}1dz$.

How I am supposed to find the other two integrals?

Finding volume
Answered

tamkieuqf
2022-08-09

I'm trying to find the volume of the solid generated by revolving the region bounded by $y={x}^{2}$ and $y=6x+7$ about x-axis using the shell method. I applied the method and I got 15864/5 multiplied by $\pi $ but it's not correct.

Details: I integrated ${\int}_{1}^{49}y(\sqrt{y}-\frac{y-7}{6})dy$