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Recent questions in Finding volume
Nina Bean 2022-08-13

Volume integral - finding the regionI 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?

heelallev5 2022-08-13

Finding value to calculate volumeGiven $D=\left[\left(x;y\right)\in {\mathbb{R}}^{2}:1\le y\le a{x}^{2}+1,0\le x\le 2/a\right],0 let W be the region obtained by rotating D around Y axis.A) Find the volume of WB) Find, if possible, the a values $ϵ\left(0;+\mathrm{\infty }\right)$ so that the volume of W is a minimum, and a maximumC) Find, if possible, the a values $ϵ\left(1/3;3\right)$ so that the volume of W is a minimum, and a maximumWell, 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 ${\int }_{1}^{\frac{4+a}{a}}\left(\sqrt{\frac{y-1}{a}}{\right)}^{2}-\left(\frac{4+a}{a}{\right)}^{2}\phantom{\rule{thinmathspace}{0ex}}dx$.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?

Cheyanne Jefferson 2022-08-13

Finding the height and top radius of cone so that volume is maximum and Finding the angle so that the volume is maximumYou 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 heighta) 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.

Jenny Stafford 2022-08-12

Finding the volume of the tetrahedron.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).

Elisabeth Wiley 2022-08-12

Finding the volume of a solid of revolution about the x-axisLet R be the area laying beneath the curve $f\left(x\right)=8-{x}^{2}$ 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.

Passafaromx 2022-08-12

Volume of Solid Enclosed by an EquationI'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 by1. $\frac{x}{a}+\frac{y}{b}+\frac{z}{c}=1,x=0,y=o,z=0$2. ${x}^{2}+{y}^{2}-2ax=0,z=0,{x}^{2}+{y}^{2}={z}^{2}$

Matonya 2022-08-12

Volume between under parabaloidSuppose 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...

daniellex0x0xto 2022-08-11

Finding volume of solid revolution about x-axisI'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 $x=-0.5$

vroos5p 2022-08-11

Find the volume of a solid which is bounded by the paraboloid $4z={x}^{2}+{y}^{2}$, the cone ${z}^{2}={x}^{2}+{y}^{2}$ and the cylinder ${x}^{2}+{y}^{2}=2x$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: to to to .For finding the volume bounded by the paraboloid and the cylinder,Bounds of integration: to to to .

dredyue 2022-08-11

How can I find the volume of a solid of revolutionFind 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$\pi {\int }_{0}^{1}\left[\left(x{\right)}^{2}-\left(\sqrt{x}{\right)}^{2}\right]dx$ 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?

vrteclh 2022-08-11

Finding a volume integral in an ellipsoidI 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 $\frac{{x}^{2}}{\left(3R{\right)}^{2}}+\frac{{y}^{2}}{\left(4R{\right)}^{2}}+\frac{{z}^{2}}{\left(5R{\right)}^{2}}=1$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$\int {\rho }_{0}\left(1-\left(9{u}^{2}+16{w}^{2}+25{w}^{2}\right)\right)60dudvdw$Does anyone have any insight to a more elegant way.

rivasguss9 2022-08-11

Finding volume of a shape using double integralI'm trying to find the volume of a given shape:using double integral. Unfortunately I don't know how to start, namely:$z=\left(1-\sqrt{y}-\sqrt{x}{\right)}^{2}$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?

musicbachv7 2022-08-10

Finding the volume of circular cone.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 $x=\left(2/3\right)y$. Am I right?

Bobby Mitchell 2022-08-10

Revolving a function around y-axis, finding volume using shell methodThe 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.

makeupwn 2022-08-10