Paraboloid Definition and 83 Threads

1. Problem: A fluid has density 1000 km/m3 and flows with velocity V =<x,y,z>, where x, y, and z are measured in meters, and the components of V are measured in meters per second. Find the rate of flow outward through the part of the paraboloid z = 64 - x^2 - y^2 that lies above the xy plane...

Homework Statement The solid enclosed by the paraboloid x=y2+z2 and the plane x=16. Homework Equations Triple integral in ractangular coordinates The Attempt at a Solution I figured out this is a paraboloid that circles the x axis, that starts at the origin and it gets wider and...

Homework Statement z = 2y^2 - x^2 Homework Equations The Attempt at a Solution I kind of know how to do it. z = y^2/b^2 - x^2/a^2 the first power is the axis of paraboloid. let x = k thus z = 2y^2 - k^2 and the vertex of this parabola (if x = 0 we see it is a...

Homework Statement Verify Stokes Theorem ∬(∇xF).N dA where surface S is the paraboloid z = 0.5(x^2 + y^2) bound by the plane z=2, Cis its boundary, and the vector field F = 3yi - xzj + yzk. The Attempt at a Solution I had found (∇xF) = (z+x)i + (-z-3)k r = [u, v, 0.5(u^2 + v^2)]...

Hi In this queation which equation we use X^2 = 4ay or y^2 = 4ax and why ? A paraboloid is formed by revolving a parabola about its axis . A spotlight in the form of a parabolid 5 inches deep has it's foucs 2 inches from the vertex . Find to one decimal place , the raduis R of the...

Homework Statement Find the volume between 2 paraboloids. Homework Equations z = 4x^2+8y^2 z= 30-x^2-y^2 The Attempt at a Solution So I switched the variables to polar coordinates. z = 30-r^2 z=4r^2*sin^2(θ)+8r^2*cos^2(θ) Now I want to solve for r. However I get a very...

Homework Statement What would the electrical resistance of a paraboloid from y = 0 to L be? Homework Equations R = \rho \frac{L}{A} The Attempt at a Solution Okay, so I'll put the parabola (that would rotate into the paraboloid) into the form y = \sqrt{x} The function A(x) is...

Homework Statement Hi. I'm asked to find the volume of the solid bounded by the paraboloid 4z=x^2 + y^2 and the plane z=4 I have drawn the graph in 3D but I'm unsure of how to set up the integral. Also, how does one decide to use double integrals/triple integrals when finding volume?

Ok its a simple question really... say that I have to find the volume (using polar coordinates) of the solid under the paraboloid z=x^2+y^2 and above the disk x^2+y^2≤9. My approach would be to find the z value of where the cylinder and paraboloid intersect. Then find the volume of the...

I've posted on this before and have now realized I was doing it completely wrong before but's still bugging me. I have to find the area of the hyperbolic paraboloid z=xy contained within the cylinder x^{2}+y^{2}=1. I've parametrized x^{2}+y^{2}=1 into polar coordinates to give that...

I need to find the area of the hyperbolic paraboloid z=xy contained within the cylinder x^2+y^2=1. I know I need to take a double integral but am having real difficulty finding the correct limits, so far I've got that; \int dx\int dy With the x limits being 1 and -1 and the upper y limit...

Homework Statement A sphere with a radius of 4 is dropped into a paraboloid. How far is the bottom point of the sphere from the bottom point of the paraboloid when the sphere stops falling? What is the radius of the largest sphere that will fall all of the way and touch the bottom of the...

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In the Schwarzschild Metric, the curvature of space around the gravitating mass can be described by the Flamm Paraboloid: w(r) = 2 \sqrt{r_{s} (r - r_{s})} Unlike the Newtonian depiction of Gravitational Potential Wells (U = - G M / r) which decrease inwards, the Flamm Paraboloid increases...

Homework Statement Find \int\int\int y^2 z^2where E is the region bounded by the paraboloid x = 1 - y22 - z2 and the plane x = 0. The Attempt at a Solution The region is a paraboloid with vertex at x = 1, y = 0, z = 0. I chose z bounds to be between 0 and 1 - y22 - z2 for first integral...

Homework Statement Parametrize the intersection of the paraboloid z = x2 + y2 and the plane 3x -7y + z = 4 between 0 \leq t \geq 2*pi When t = 0, x will be greatest on the curve.Homework Equations The Attempt at a Solution I never really know how to do these kinds of problem. I am more...

Homework Statement evaluate volume of paraboloid z = x2 + y2 between the planes z=0 and z=1 The Attempt at a Solution i figured we would need to rearrange so that F(x,y,z) = x2 + y2 - z then do a triple integral dxdydz of the function F. the limits for the first integral dz would be z=1...

Homework Statement Given the elliptic paraboloid of height H and two semiaxes A and B. How to find its surface area? Homework Equations x = A * sqrt(u) * cos(v) y = B * sqrt(u) * sin(v) z = u u belongs to [0; H], v belongs to [0; 2*PI)

Homework Statement So my question is: what is the volume of the region R between the paraboloid 4-x^2-y^2 and the xy-plane? Homework Equations I know how to solve it, it is a triple integral, but how do you find the limits of integration? The Attempt at a Solution Do I set x=0...

Greetings again, this is another I'm stuck on... Calculate the volume under the elliptic paraboloid z = x^{2} + 4y^{2} and over the rectangle R = [-1, 1] \times [-4, 4]. I'm not sure how to find the limits of z for the triple integral. Can I somehow integrate the function z, and only...

Homework Statement Consider the parametric surface r(u,v)=<vsinu, vcosu, v^2> a) Identify the shape of the surface b) The point (1,1,2) is on the surface. Find: i) A grid curve wit hv constant that contains this point ii) A grid curve with u constant that contains this point c)...

Lately I've been trying to teach myself GR and it's been going fairly well, but yesterday for practice I decided to compute the curvature tensor of a paraboloid and it's not working. I've tried using three different coordinate systems, starting with what I thought would be the most obvious one...

Homework Statement Particles are scattered (classically) from a paraboloid shape. The surface is given by the relation: z = a \left(\frac{y^2+x^2}{R^2}-1 \right) for x^2 +y^2 leq R^2 where a and R are constants. The particle is incident from z = -infinity with impact parameter s. Show that...

Homework Statement A body of mass M moves (in a gravitational field g) on the inner surface of given by equation: z=\frac{1}{2a}(x^{2}+y^{2}) (a is positive) Reduce the question of finding the motion to quadratures. Homework Equations The Attempt at a Solution I used...

Evaluate the volume of the solid bounded by the plane z=x and the paraboloid z = x^2 + y^2 I have tried to graph this, and they don't bound anything? have i graphed it wrong. and is there a way to do these problems where you don't need to draw the graph.

Geodesics of hyperbolic paraboloid (urgent!) Help me find the geodesics of the hyperbolic paraboloid z=xy passing through (0,0,0). I know that lines and normal sections are geodesics. Based on a picture, I think y=x and y=-x are 2 line geodesics. Then, maybe the planes in the z-y and z-x...

Homework Statement I am wondering what the set of area vectors for a surface would be. For a plane on the xy-plane, I know the set of area vectors is <0,0,dx*dy>. Homework Equations So, for a set of points (x,y,z) that make a paraboloid, if F(x,y,z)=0 then [grad(F)•<dy*dz, dx*dz...

I have a classical mechanics question I couldn't conclude. The reason seems to be mathematical. It's this: There's a paraboloid shaped plane of mass M, which is standing on a frictionless surface and can slide freely. It's surface is y=ax^2. A point mass m is place on the plane. Solve the...

Question: Consider the intersection of the paraboloid z = x^2 + y^2 with the plane x - 2y = 0. Find a parametrization of the curve of intersection and verify that it lies in each surface. How I went about it: x = 2y z = (2y)^2 + y^2 = 5y^2 Set y = t, then x = 2t y = t z = 5t^2...

I need to find a set of parametric equations for a hyperbolic paraboloid. The hint is that I should review some trigonometric identities that involve differences of squares that equal 1. The equation is: \frac{y^2}{2}- \frac{x^2}{4} - \frac{z^2}{9} = 1 And what I have is...

how do I parametrize the paraboloid z = x^2 + y^2 ? thx

Original question: a) Find a vector function for the curve of intersection of the paraboloid z = 3x^2 + 2y^2 and the cylinder y = x^2. b) Show that this curve passes through (1,1,5) but not (3,3,9). I really have no idea how to do either parts of this question. Any help would be greatly...

Hi, I'm really stuck on this problem and I need some help?? Here's the question: The intersection of the elliptic paraboloid z=x^2+4y^2 and the right circular cylinder x^2+y^2=1. Use Lagrange multipliers to find the highest and lowest points on the curve of intersection. Your help will...