Paraboloid Definition and 81 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...

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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...

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...