Triple integral Definition and 316 Threads

Homework Statement Find the volume of the solid in the octant x,y,z>=0, bounded by x+y+1=1 and x+y+2z=1 The Attempt at a Solution I've been looking at an example in the textbook that is similar to this problem. First, I found the projection of W onto the xy plane...

Homework Statement Evaluate the triple integral for specified function and box B. f(x,y,z) = x ey-2z 0<x<2, 0<y, z>1 (The < and >'s should be less than or equal to but, I don't know how to write that here) Homework Equations The Attempt at a Solution I know how to evaluate...

Homework Statement Homework Equations The Attempt at a Solution

Homework Statement This is a question an exam I already took but I felt as though the solution posted by the professor might be incorrect so I wanted to hear some other opinions. The question is as follows: What is the maximum value that \iiint (-|x|-|y|+1)\,dV will compute for any region...

I'm trying to set up this triple integral with the following bounds: x=0, y=0, z=0, x+y=1, z=x+y. Now I first computed the volume to be 1/3 with a double integral and then what I've been doing is setting what I think are the right bounds for the triple integral and integrating f(x,y,z)=1...

find the triple integral of xy where E is bounded by y = x^2 and x = y^2 and the planes z = 0 and z = x + y. i got 1/3 as a solution, but I'm not sure if i did it right, specifically the part in finding the boundaries for x and y. i found that they intersected at (0,0) and (1,1) so i had the...

Homework Statement Find the volume of the region which is bounded by the parabolic cylinders y=x², x=y² and z=x+y and z=0 Homework Equations The Attempt at a Solution I solved x=y² for y, and set that equal to y=x², and I got the intersection of the two parabolic cylinders to...

Homework Statement \int^{1}_{-1}\int^{1}_{x^2}\int^{1-y}_{0} dz dy dx Homework Equations See the attachment for graph. I am supposed to rewrite the order of integration to the following. a)dy dz dx b)dy dx dz c)dx dy dz ...and so on. The Attempt at a Solution First attempt is...

Homework Statement use triple integral to find the volume of tetrahedron enclosed by the coordinat planes "x=o , y=0 , z=0" and the plane 2x+y+z=0 Homework Equations The Attempt at a Solution I will integrate the constant function f(x,y,z)=1 by the order : dzdydx the...

Homework Statement Use the Divergence Theorme to evaluate the flux of v(x,y,z)=x^{2}i+y^{2}j+z^{2}k on the solid T bounded above by a sphere with radius 3 and below by the xy-plane. I've found that div(v) is 2(x+y+z). When I go to set up the integral I get a triple integral over T of...

Homework Statement Use cylindrical coordinates to find the volume of the solid. The solid is enclosed by the paraboloid z=x2+y2 and the plane z=9Homework Equations z=r2 The Attempt at a Solution So I'm getting close to the answer but not quite, and I keep getting a negative which doesn't make...

Homework Statement Use a triple integral to find the volume of the solid. The solid in the first octant bounded by the coordinate planes and the plane 3x+6y+4z=12 Homework Equations z=3-\frac{3}{4}x-\frac{3}{2}y The Attempt at a Solution So I'm using the triple integral...

Homework Statement Evaluate \int\int\int 1/\sqrt{x^{2}+y^{2}+z^{2}+3} over boundary B, where B is the ball of radius 2 centered at the origin. Homework Equations Using spherical coordinates: x=psin\Phicos\Theta y=psin\Phisin\Theta z=pcos\Phi Integral limits: dp - [0,2] d\Phi -...

Homework Statement use a triple integral to find the volume of the region that is common to the interiors of z^2 +y^2 + z^2 = 1 and x^2 + z^2 = 1 Homework Equations Would I just calculate the are of the disc? I set up a triple integral as inte [0 to 1] 2nd inte [0 to sqrt(1-z^2)] 3rd...

Homework Statement The region R in 3D is cut from the first octant (x,y,z >= 0) by the plane X+Z = 1, Y+2Z = 2. Set up the volume in all 6 ways in rectangular coordinates. Then evalute the volume in two of these ways. Make sure to specify limits of integration in every case. Homework...

Use cylindrical coordinates to evaluate the triple integral , sqrt(x^2+y^2) where the region integrated is the solid bounded by the circular paraboloid z=9-16(x^2+y^2) and the xy-plane. I'm having trouble deciding what the bounds for r would be.

Homework Statement Find the volume of the solid which is contained by 1) z= \frac{\sqrt{2}}{4}\sqrt{x^2+y^2} and 2) x^2+y^2+z^2= \sqrt{27}z Homework Equations I've completed the square on the 2nd equation to obtain x^2+y^2= 8z^2 and also the 1st equation to obtain...

Homework Statement Show that \int^{infinity}_{-infinity}\int^{infinity}_{-infinity}\int^{infinity}_{-infinity}sqrt(x^2+y^2+z^2)e^-^(^x^2^+^y^2^+^z^2^)dxdydz = 2\pi Homework Equations x^2+y^2+z^2 = \rho^2 The Attempt at a Solution I converted to spherical coordinates to get...

Homework Statement Use spherical coordinates to find the volume of the solid that lies above the cone z2 = x2 + y2 and below the sphere x2 + y2 + z2 = z Homework Equations I'm going to use { as an integral sign. Volume = {{{ P2 Sin[Φ] dP dΦ dΘ The Attempt at a Solution P2 =...

Homework Statement A conical container with radius 1, height 2 and with its base centred on the ground at the origin contains food. The density of the food at any given point is given by D(r) = a/(z + 1) where a is a constant and z is the height above the base. Using cylindrical polar...

Homework Statement Find the volume of the solid bounded by the cylinder x^2+y^2=9 and the planes y+z=5 and z=1Homework Equations NoneThe Attempt at a Solution My main problem is setting up the integral. So far what I have is 1 as the integrand, my order of integration is dydxdz and my bounds...

Homework Statement Inside the sphere x2 + y2 + z2 = R2 and between the planes z = \frac{R}{2} and z = R. Show in cylindrical and spherical coordinates. Homework Equations \iiint\limits_Gr\,dz\,dr\,d\theta \iiint\limits_G\rho^{2}sin\,\theta\,d\rho\,d\phi\,d\theta The Attempt at a...

Hi I have a homework set due this week, 14 problems, I have done 11 of them, but these 3 are giving me trouble, help would be great :) Homework Statement 1.A cylindrical drill with radius 4 is used to bore a hole through the center of a sphere of radius 8. Find the volume of the ring shaped...

Homework Statement Use cylindrical coordinates to evaluate the triple integral (over E) sqrt(x^2+y^2) dV , where E is the solid bounded by the circular paraboloid z=9−16(x^2+y^2) and the xy plane. The Attempt at a Solution This is really bugging me... Is this the correct setup for the...

Homework Statement Find the region E for which the triple integral: (triple integral over E) (1 - x^2 -2y^2 -3z^2) dV is a maximum. Homework Equations The Attempt at a Solution I remember in earlier math courses finding the derivative of a single variable integral, does this...

hi all how can i find the volume of the solid that lies within the sphere x^2+y^2+z^2=36 , above the xy plane, and outside the cone z=7sqrt(x^2+y^2) . your help is very much appreiated

Homework Statement Let S be the region in the first octant under the plane 3x + 2y +z = 4. Find the volume of S. Homework Equations idk? The Attempt at a Solution \int^{\frac{4}{3}}_{0}\int^{\frac{3}{2}x + 2}_{0}\int^{-3x - 2y + 4}_{0}dzdydx =...

[b]1. consider the triple integral (x^2 +Y^2) dV where it is bounded by a solid sphere of radius R. Set up the integral using rectangular coordinatesI tried setting this up with the bounds [ -sqrt(R^2-x^2-Y^2) <= Z <= sqrt(R^2-x^2-Y^2) , -R <= X <= R , -sqrt(R^2-x^2) <= Y <= sqrt(R^2-x^2) ]...

Homework Statement Given that: Write an equivalent integral in spherical coordinates. Homework Equations (Triple integral in spherical coordinates.) (Conversions from rectangular to spherical coordinates.)(What spherical coordinates entail) The Attempt at a Solution The region...

I'm taking a Calculus class as an elective. This might not have been a good idea, but I'm stuck in it now. Here is a problem I have to do. My knowledge of basic maths is poor, so please be gentle and explain thoroughly! 1. The problem. Rewrite the following integral in terms of spherical polar...

So I'm in the middle of a calculus 3 course, and one thing I've been lightly chewing on is how to change the direction of evaluation of a double/triple integral when the bounds are complicated enough that they can't be drawn easily on a graph. Would you have to use the optimization in several...

HELP! Setting up triple integral in spherical coordinate Homework Statement http://img517.imageshack.us/img517/9139/83291277.jpg Homework Equations I set up the bound for this problem as following: r=0..2/cos(phi), phi=pi/2..3pi/4, theta=0..2pi, but maple always return an error in...

Homework Statement I need to set up the triple integral to find the volume of the region bounded by the sphere x2 + y2 + z2 = a2 and the ellipsoid \frac{x^2}{4a^2} + \frac{4y^2}{a^2} + \frac{9z^2}{a^2} = 1 Homework Equations The Attempt at a Solution I solved it in spherical...

Homework Statement Use a triple integral in rectangular coordinates to find the volume of the ice cream cone defined as follows The region R in the xy-plane is the circle of radius 1 with center at the origin. The sides of the cone are defined by the function z= \sqrt{x^2+y&2} The top of...

Homework Statement I need to set up the triple integral to find the volume of the region bounded by the sphere: x^2+y^2+z^2=a^2 and the ellipsoid: (x^2/4a^2)+(4y^2/a^2)+(9z^2/a^2)=1. Homework Equations above The Attempt at a Solution I'm not sure which interval I should be using here. I...

Homework Statement Evaluate this integral using spherical coordinates: http://img262.imageshack.us/img262/9361/xyzm.th.jpg Homework Equations http://img40.imageshack.us/img40/9508/conss.th.jpg The Attempt at a Solution http://img264.imageshack.us/img264/9457/attempt.th.jpg...

Triple Integral ( IS THIS RIGHT??) Homework Statement Let R be the solid enclosed by the planes x=0, y=0, z=2 and the surface x^2+y^2, where x\geq0, y\geq0. Compute\int\int\intxdxdydz Homework Equations The Attempt at a Solution I did \int0-1\int0-1\int(x^2+y^2)-2 xdzdydx...

Homework Statement Use cylindrical coordinates to evaluate the triple integral \int\int\int \sqrt{x^2+y^2} dV in region E where E is the solid bounded by the circular paraboloid z=9-(x^2+y^2) and the xy-plane. Homework Equations knowing that x = rcos\theta y= rsin\theta z=z...

Just needing to check an answer really. The question is as follows; The domain bounded by the surface of a paraboloid z=2-x^2-y^2 and that of a cone z^2=x^2+y^2 is given by D = ( x,y,z : x^2+y^2 \leq 1, sqrt(x^2+y^2) \leq z \leq 2-x^2-y^2 ). Find its volume using the appropriate coordinate...

How would I evaluate the triple integral \int\int\int_H(x^2+y^2) dV, where H is the region bounded x2 + y2 = 1, y = x, y = 0, z = 0, z = 2

How would I evaluate the integral \int\int\int_{G} \sqrt{4x^2+9y^2} dV, where G is the elliptic cylinder 4x2+9y2 \leq 25, 0 \leq z \leq 6

Given the triple integral \int\int\int_{G} xyz dV Where G is the region bounded by x=1, y=x, y=0, z=0, z=2. How do I evaluate it. Please help.

Homework Statement Evaluate the triple integral where E is the solid bounded by the cylinder y^2+ z^2 = 576 and the planes x = 0, y = 4 x and z =0 in the first octant. Homework Equations The Attempt at a Solution I figure that by solving for z I can get the bounds, so between 0 to...

If \int\int\int_{Q}dV = Volume_{Q}, and graphically, it represents the volume between all the boundaries of the respective variables in the iterated integral, what does \int\int\int_{Q}f(x,y,z)dV represent? Does this integral represent the (volume?) "above" (in 4D sense) the solid represented by...

Homework Statement A container has a vertical height of 1m, a circular base with radius 1m and a circular top with radius 2m. Use a triple integral and spherical coordinates to evaluate the volume of the container. 2. The attempt at a solution If we set up the problem so that the centre of...

Homework Statement Evaluate the following triple integral I = \int\int\int_{R}x dv in Cartesian coordinates where R is the finite region bounded by the surfaces z=0, y=x^3, y=8, z=x. Sketch the region R. Here dV is the element of volume. Homework Equations The Attempt at a...

Homework Statement Find \int \int \int_{D} xydV, where D is the solid bounded by the coordinate planes, the plane x = 1 and the surface z = 16 - 4x^2 - y^2. Homework Equations The Attempt at a Solution I have no problem with actually performing the integration, but I'm lost on...

Find the volume of the region above the sphere x^2+y^2+z^2=4 and below the paraboloid z = 6-x^2- y^2. Ok so the first thing i did was to find out if the two surfaces ever intersect by substituting x^2+y^2=6-z into the first equation and solving for z. I got only complex solutions, hence they...

Homework Statement evaluate ysinx from z= 0 to (1-y2)1/2, y = 0 to 1, x = 0 to pi Homework Equations The Attempt at a Solution heres my work -1/2sinx \int(1-y^2)^1/2 y dy from 0 to 1 = \int1/3*sinx dx from 0 to pi = 2/3 Homework Statement

I have general question which need to be answered before I can understand steps which I have to do. There are: When you are told that a solid is bounded by the coordinate plane and the plane x+10y + 2z = 5, are the limits considered to be 0-1 for x-axis, 0-10 for the y-axis and 0-2 for the...