Triple integral Definition and 316 Threads

Find the mass of a solid bounded by x = (4-y2)1/2 y = 0 z = 0 z = 1 + x with density = y i understand how to set it upand transform to polar and how to do it but my teacher said its supposed to be -pi/2 to pi/2 for the integral with respect to theta. shouldn't it be 0 to pi/2 because its...

Homework Statement Consider a region in the first octant bounded by z=1-x^2 and y = 1-x. Write the integral in the order dx dy dz and dx dz dy. Homework Equations The Attempt at a Solution For the first one: z varies from 0 to 1. y (in terms of z) varies from...1 to 1?? x (in terms of z...

The question states: Find the center of mass of the solid that is bounded by the hemisphere z = sqrt(21 - x ^2 - y^2) and the plane z = 0 if the density at a point P is directly proportional to the distance from the xy-plane. I know that the integral is setup : m =...

Homework Statement http://img3.imageshack.us/img3/7558/47586628.th.jpg Homework Equations The Attempt at a Solution I am quite confused whether I should use cartesian, cylindrical, and spherical coordinate.. how do I approach this problem

i am not using the template because it doesn't really apply to my question. can anyone explain to me when to use double integral or triple integral for volume? what clues should i look for in the question? also i am still unsure when are polar coordinates used as opposed to cylindrical and...

Homework Statement http://img5.imageshack.us/img5/5222/53026504.th.jpg Homework Equations The Attempt at a Solution I know A-F except for what E is here, I answered sqrt(x^2+y^2) but it is wrong, so what is it supposed to be?

Homework Statement http://img5.imageshack.us/img5/6596/67023499.th.jpg http://img5.imageshack.us/img5/3875/13930604.jpg Homework Equations The Attempt at a Solution my answer to the first one is negative and the second one is positive, is this correct?

Homework Statement http://img12.imageshack.us/img12/7181/integral.th.jpg Homework Equations The Attempt at a Solution Well my first attempt is to convert this to a cylindrical coordinate first, which I believed to be: \int_0^1 \int_0^{2\pi} \int_0^1 1 \, dr \,d\theta \,dz is this correct?

Homework Statement http://img19.imageshack.us/img19/5192/captureonr.th.jpg Can anyone tell me if I did any mis-computation on evaluating the triple integral above? Homework Equations The Attempt at a Solution

Homework Statement http://img19.imageshack.us/img19/2559/triple.th.jpg Homework Equations The Attempt at a Solution I get 16pi/3 (sqrt(2) -1) as the final result, but when I input the answer to the computer, it doesn't accept it. Am I doing a wrong integration/calculation...

Homework Statement The figure below shows part of a spherical ball of radius 5 cm. Write an iterated triple integral which represents the volume of this region. http://img19.imageshack.us/img19/9237/sphereu.th.jpg http://img19.imageshack.us/img19/1699/inte.jpg Homework Equations The Attempt...

SOLVEDHomework Statement evaluate the intergralHomework Equationssorry about how this is going to look don't know the language to display nicely and wouldn't take my copy and pasteall integrals are form -infinity to infinity (x^2+32*z^2)*cos(y)*e^(x-4*z) delta(x-1) delta (y-pi) delta(z-.25)...

Homework Statement A triple integral, with the bounds, from outer to inner: integrate from -1 to 1 with respect to x integrate from 0 to 1-x^2 with respect to y integrate from 0 sqrt (y) with respect to z on the function x^2*y^2*z^2Homework Equations noneThe Attempt at a Solution I know what...

Homework Statement set up an integral in cylindrical coords to compute the volume of the solid S bounded by the sphere x^2+y^2+z^2=12 and the cone 3z^2=x^2+y^2 where z>=0 The Attempt at a Solution i will post my answer here. please let 'I' stand for integral: i get, I[0,2pi]...

Homework Statement Essentially, do the volume integral of z^2 over the tetrahedron with vetices at (0,0,0) (1,0,0) (0,1,0) (0,0,1) The Attempt at a Solution There seems to be a ton(!) of brute-force algebra involved. Enough to make me question if I'm doing the problem right. I set up the...

Homework Statement \int\int_{Q}\int(x^4+2x^2y^2+y^4)dV where Q is the cylindrical solid given by \{(x,y,x)| x^2+y^2 \leq a^2, 0\leqz\leq\frac{1}{\pi}\}Homework Equations When I convert to cylindrical I get f(r,\theta,z) = r^4\cos^2\theta + 2r^4\cos^2\theta\sin^2\theta + r^2\sin^2\theta, but I...

Homework Statement Use spherical coordinates to find the volume of the solid bounded above by the sphere with radius 4 and below by the cone z=(x^2 + y^2)^(1/2).Homework Equations All general spherical conversions Cone should be \phi=\pi/4The Attempt at a Solution So far I think the triple...

Homework Statement W is the solid bounded by the three coordinate planes and the surface 4x+2y+3z=16, Calculate Mxz=\int\int\int y dV Homework Equations The Attempt at a Solution the surface 4x +2y +3z=16 is a plane that crosses boundries at (4,0,0), (0,8,0) and (0,0,16/3)...

\int \int \int cos(u + v + w)dudvdw (all integrals go from 0 to pi). I've tried using u substitution for each integral but I end up with a huge integral.

Homework Statement This is my last question about triple integrals in cylindrical coordinates. Evaluate the integral by changing to cylindrical coordinates: \int _{-3}^3\int _0^{\sqrt{9-x^2}}\int _0^{9-x^2-y^2}\sqrt{x^2+y^2}dzdydx Homework Equations In cylindrical coordinates...

Homework Statement Find the mass and center of mass of the solid S bounded by the paraboloid z=4x^2+4y^2 and the plane z=a\;\;(a>0) if S has constant density K. Homework Equations In cylindrical coordinates, x^2+y^2=r^2. The Attempt at a Solution In order to find the mass, I tried...

Homework Statement Evaluate \int \int \int_E x^2 \, dV where E is the solid that lies within the cylinder x^2+y^2=1, above the plane z=0, and below the cone z^2=4x^2+4y^2.Homework Equations In cylindrical coordinates, x^2+y^2=r^2 and x=r\cos{\theta}.The Attempt at a Solution I tried \int...

Homework Statement Use a triple integral to find the volume of the solid enclosed by the paraboloid x=y^2+z^2 and the plane x=16 Note: The triple integral must be performed in Cartesian coordinates. Homework EquationsThe Attempt at a Solution I calculated the answer numerically using...

Revised question is below.

I don't think so since it's not a sphere (disk). I have not learned about cylindrical coordinates and Cartesian is just a pain, so I am assuming I am supposed to use polar or something. Can someone clear up my confusion? \int\int\int_E y\,dV where E lies above the plane z=0, under the plane...

gravity inside a solid sphere Homework Statement I'm having a hard time setting up a triple integral to find the force of gravity inside a solid sphere. I've done a similar proof in physics before with gravity inside a spherical shell, but it only required a single integral. In this problem...

Homework Statement I=\int\int\int_E x^2e^ydV where E is bounded by the parabolic cylinder z=1-y^2 and the planes z=0 x=1 and x=-1 I know that the graph is a parabola that opens downwards and that has symmetry wrt the x-axis. It also stretches along the x-axis toward + and - infinity...

Homework Statement Integrate the function over the solid given by the figure below (the bounding shapes are planes perpendicular to the x-y plane, a cone centered about the positive z-axis with vertex at the origin, and a sphere centered at the origin), if P=(0,0,5),Q=(0,4,3), and...

Homework Statement Integrate the function over the solid given by the "slice" of an ice-cream cone in the first octant bounded by the planes x=0 and and contained in a sphere centered at the origin with radius 20 and a cone opening upwards from the origin with top radius 16...

Alright guys I am looking for some help with this problem regarding calculating total electric charge in a layer of ions. This layer of ions is bounded between the planes x+2y+2z=4 and x+3y+3z=3, and by the 3 co-ordinate planes. The density of the ions is rises linearly from zero at the outer...

let be the integral \int_{R^{3}}d^{3}r F( \vec r . \vec r , \vec r . \vec a , |r| ,|a|) (1) F depends only on the scalar product of vector r=(x,y,z) and its modulus |r| , hence it is invariant under rotation and traslations (since scalar product is invariant under rotation and...

Homework Statement Triple integral of 1+z inside the cone z=2sqrt(x^2+y^2) above the xy plane and bounded by z=6 Homework Equations The Attempt at a Solution when z=6, 6=2sqrt(r^2) so r=3 limits of integration are z=6 to z=2r r=3 to r=-3 theta=2pi to theta=0 Just want to make...

Homework Statement Find the triple integral of z where E is bounded by the planes z=0 y=0 x+y=2 and the cylinder z^2+y^2=1 in the first octant. Homework Equations The Attempt at a Solution Just want to make sure that my setup is right. The limits of integration of x are 2 to 0, for z...

I thought this question was elementary... but I apparently know less than I thought I did. Homework Statement Use spherical coordinates to evaluate \iiint_{E} x^{2}+y^{2}+z^{2}dV Where E is the ball x^{2}+y^{2}+z^{2}\leq 16 Homework Equations x^{2}+y^{2}+z^{2}=\rho^{2} The...

Homework Statement Find the centroid of the solid: the tetrahedron in the first octant enclosed by the coordinate planes and the plane x+y+z=1. Homework Equations xcenter = \frac{\int\int\int_G x dV}{V} ycenter = \frac{\int\int\int_G y dV}{V} zcenter = \frac{\int\int\int_G z dV}{V} The...

Homework Statement Find \int\int\int\sqrt{x^2+y^2+z^2}e^{-x^2-y^2-z^2}dxdydz The limits of integration for all 3 variables are from -infinity to infinity. Homework Equations This one has me completely stumped, so I'm just wondering if someone could push me in the right direction in how...

hi everyone the integral is : \[ I = \int\limits_0^1 {\int\limits_0^x {\int\limits_0^y {ydzdydx} } } \] I'm not sure about the answer , but i think it'll be \[ \frac{{x^3 }}{3} \] am i right ? thanks

Homework Statement Find the triple integrals \oint\oint\oint_{W}{f(x,y,z)dV: e^{x^{2}+y^{2}+z}, (x^{2}+y^{2}) \leq z \leq {(x^{2}+y^{2}})^{1/2}Homework EquationsThe Attempt at a Solution So I know I need to probably switch to cylindrical coordinates. But I'm getting confused about the limits...

[SOLVED] Triple integral Homework Statement Calculate the triple integral of z when z [(r-1), sqrt(1-(r-2)^2)], r [1, 2], tetha [0, 2*pi] 2. The attempt at a solution I've tried again and again, and I always get (17/4)*pi, while the answer is pi/2. Is there anything wrong with...

I'm supposed to find the volume of a solid bound by co-ordinate planes x=0,y=0, z=0 & surface z=1-y-x^2 and am having a lot of difficulty doing so. f(x,y,z) is not given so I am assuming it is one. I figure I should then take the triple integral dzdydx. Then, I made a 2D sketch for the xy plane...

Homework Statement Evaluate the following integral: \iiint \,x\,y\,z\,dV Where the boundaries are given by a sphere in the first octant with radius 2. The question asks for this to be done using rectangular, spherical, and cylindrical coordinates. I did this fairly easily...

Triple Integral Help :( Can anyone help me with this triple integral problem? I'm sorry I don't know how to post the script properly; I'm a complete newb. It's a surface integral problem- that part is not important- I have to calculate a triple integral where S is the surface of the volume...

For my assignment I have to come up with a really complex double or triple integral that equals 30. Would you mind giving me some ideas?

Homework Statement So my question is as follows: Find the volume of the solid bounded by z = 3x^2 + 3y^2 and z = 6sqrt(x^2 + y^2) The Attempt at a Solution I drew the graphs of these out, with the z = 3x^2 + 3y^2 being a circular paraboloid w/ vertex at (0,0,0) and extending in...

Homework Statement I am to derive the volume of a sphere using spherical coordinates. I have derived the (correct) jacobian as r^2sin(theta) dr d(theta) d(phi) so its simply a matter of integrating over the correct limits. Homework Equations What I don't get is why we use 2pi to 0 for...

Triple Integral Evaluation (quick and easy) Homework Statement \int_{0}^{1} \int_{x^2}^{1} \int_{0}^{3y} ({y+2x^2z})dz dy dx Homework Equations None. The Attempt at a Solution Here is what I got at the end (the LaTeX takes too long to code in here, plus its not showing up)...

Hi, my result of \int \int \int_{A} xyz dxdydz where A = \{(x,y,z); x^2+y^2+z^2 \leq 2, x \geq 0, y \geq 0, z \geq 0 \} is \frac {8}{48} , but book says \frac{1}{48} . Is the book right? Could you please verify? Thank you Michael

Integrate the function f(x,y,z)=–6x+2y over the solid given by the "slice" of an ice-cream cone in the first octant bounded by the planes x=0 and y=sqrt((277/123))x and contained in a sphere centered at the origin with radius 25 and a cone opening upwards from the origin with top radius 20. I...

Find the volume of the solid inside the cylinder x^2+y^2=2x and bounded by the sphere x^2+y^2+z^2=4 It appears that cyclindrical is out the question because there is no symmetry about the centre of the cylinder. So only spherical coords are applicable. Any clues?

I like to use cartesian coords Find the region to the intersecting cyclinders x^2+y^2<=a^2 and x^2+z^2<=a^2 What I have trouble finding is the domain of integration Currently I have a to -a for dx -srt(a^2-x^2) to srt(a^2-x^2) for dy -srt(x^2+y^2) to srt(x^2+y^2) for dz But this...