Triple integration Definition and 34 Threads

Homework Statement If ##\vec { F } = x \hat { i } + y \hat { j } + z \hat { k }## then find the value of ##\int \int _ { S } \vec { F } \cdot \hat { n } d s## where S is the sphere ##x ^ { 2 } + y ^ { 2 } + z ^ { 2 } = 4##. The Attempt at a Solution From gauss divergence theorem we know ##\int...

Homework Statement Homework Equations in the pic The Attempt at a Solution why is there an extra "r" in the highlighted line? my attempt: ## \int_0^{2\pi}\int_0^2\int_{r^2}^41\ dy\ r\ dr\ d\theta ## = ##\int_0^{2\pi}\int_0^2\left(4-r^2\right)\ r\ dr\ d\theta## thanks

Homework Statement I'm trying to figure out the other parameters to solve the problem Homework Equations I know sqrt(x^2+y^2+z^2) = p The Attempt at a Solution I changed the integrand to p^3 sin(theta) since p * p^2 sin(theta) Then for the first integration sign, I know how to get the...

Homework Statement Shown in the photo attached. 2. Homework Equations ∫V r2Sinθdθdφdr in spherical coordinates ∫V dxdydz in cartesian coordinates equation of a sphere x2+y2+z2=r2 The Attempt at a Solution In this case y=(y-2): sphere displaced on the y-axis. and since it is bound by all...

Homework Statement Homework Equations spherical Jacobean The Attempt at a Solution I have (sorry, have to capture my work, too hard to type) then the integration of p3 ep2 = 1/2 ep2 (p2-3/2) ??

Homework Statement \int_0^5 \int_0^2 \int_0^{4-y^2}\ \, dxdydx Change order to dydxdz Homework EquationsThe Attempt at a Solution I'm confused mainly because the limits are mostly numbers, not functions. I graphed the limits in @D and #d and this is what I got: \int_0^{4-y^2} \int_0^5...

Homework Statement The problem asks for a single triple integral (the integrand may be a sum but there must be a single definition for the bounds of the integral) representing the volume (in the first octant) of the shell defined by a sphere of radius 2 centered around the origin and a sphere...

Homework Statement Evaluate the triple integral: ∫ x dxdydz A where A = {(x; y; z) : x, y, z > 0, x + y + z ≤ 1} . Homework Equations None that I know of. The Attempt at a Solution The problem I have is determining the limits for x, y and z. I don't really understand the following...

Homework Statement Homework Equations ∫∫∫dV The Attempt at a Solution Ok so I started by setting my bounds equal to √(200-x^2-y^2) ≥ z ≥ √(x^2+y^2), √(100-x^2) ≥ y ≥ -√(100-x^2), 10 ≥ x ≥ -10 which I got from solving z^2 = (200-x^2-y^2) = x^2+y^2 => x^2+y^2 = 100 but it...

Homework Statement Express the iterated integral ∫[0,1]∫[0,1-y^2]∫[0,y] f(x,y,z)dzdxdy a. as a triple integral (i.e., describe the region of integration); b. as an iterated integral in the order z, y, x; c. as an iterated integral in the order y, z, x: The Attempt at a Solution so...

Homework Statement Convert the integral from rectangular coordinates to spherical coordinates 2 √(4-x^2) 4 ∫ ∫ ∫ x dz dy dx -2 -√(4-x^2) x^2+y^2 Homework Equations x=ρ sin∅ cosθ y=ρ sin∅ cosθ z=ρ cos∅ In case the above integrals cannot be understood: -2...

Homework Statement a)Calculate the total charge a square capacitor plate would have with width x, height y, thickness z, and charge density f(x,y,z) = 1+x+y b)Calculate the total charge a sphere would have with radius r, and charge density f(x,y,z)=x+y+z Use the triple integration seen in...

Homework Statement Find the volume of the solid in the first octant bounded by the graphs of: z=1-y2 y=2x x=3 Homework Equations I was able to graph all three but I can't picture the region for integration. I'm not sure if I even have to graph it or if I can get my limits without the graph...

Homework Statement Determine the value of \int_{0}^{1} \int_{0}^{\sqrt{1-x^2}} \int_{0}^{\sqrt{1-x^2-y^2}} \sqrt{x^2+ y^2 + z^2} dz dy dx The Attempt at a Solution So in spherical polars, the integrand is simply ρ. \sqrt{1- x^2- y^2} = z = ρ\cos\phi = \cos\phi since we are on the unit...

Homework Statement http://s2.ipicture.ru/uploads/20111231/kczcXUuF.jpg The attempt at a solution So, I'm using the transformation to spherical coordinates (ρ,∅,θ) Description of region: For θ and ∅ fixed, ρ varies from 0 to 4. For θ fixed, ∅ varies from 0 to ∏. (i suspect the error...

Homework Statement Find all 5 other orders of intergration Homework Equations \int_{0}^{1}\int_{0}^{x^2}\int_{0}^{y}dzdydx The Attempt at a Solution I am really confused as to how to represent it graphicaly so I don't have any visuals to help. Can you guys help me? Thank you SO...

Homework Statement Evaluate the integral by changing the order of the integration in an appropriate way. ∫∫∫ ze-(y2+z2)dzdydx z goes from 0 to ∞, y goes from x/6 to 3, x goes from 0 to 18 Homework Equations The Attempt at a Solution to change the integration ∫∫∫...

Find the volume lying below z = 3 - 2y and above z = x^2 + y^2. How would I go about finding the limits of integration for this problem?

Homework Statement \int _0^{\sqrt{\pi }}\int _0^x\int _0^{x z}x^2 \text{Sin}[y]dydzdx

I'm given this definite integral: \int_0^{1}\int_{\sqrt{x}}^{1}\int_{0}^{1-y}f(x,y,z)dzdydx I need to change the order to dydxdz, but I'm stuck trying to get the limits of integration wrt y. \int_0^{1}\int_{x^2}^{0}\int_{}^{}f(x,y,z)dydzdx How do I find the limits of y?

Homework Statement Rewrite \int_{0}^{2}\int_{0}^{y^3}\int_{0}^{y^2}dzdxdy as an integral with order dydzdx. Homework Equations N/A The Attempt at a Solution Honestly, I got as far as sketching it: and after sketching it, I'm lost... I can't figure out how to set z or y, but...

Homework Statement Can anyone help me with the volume of a 2-sphere in rect cordinates? I'm having problems with the limits of the triple integral. Ultimately I will need to go beyond the 2-sphere to a 3 and 4 using quadruple and five integrals respectively. Radius at r from 0 vector. Homework...

I need to find the volume between the cone z=sqrt(x^2+y^2) and the sphere x^2+y^2+z^2=1 that lies in the first octant. Now I've used cylindrical coordinates for this and found the limits to be 0<theta<pi/2 0<r<1/sqrt(2) r<z<sqrt(1-r^2) I've done the triple integral and found the answer...

Homework Statement Use spherical coordinates to evaluate the triple integral z dV where Q is the solid that lies between x^2+y^2+z^2=1 and x^2+y^2+z^2=4. Homework Equations Not sure what goes here :P The Attempt at a Solution I've gotten everything set up, I am having problems with boundaries...

Homework Statement Sketch the solid B that lies inside the surface x^2 + y^2 = 1 and is bounded above and below by the surface x^2 + y^2 + z^2= 2^2. Then find the volume of B. Homework Equations projxy = projection onto the xy plane, proj zy = projection on the zy plane The Attempt...

Homework Statement The problem was to find the volume enclosed by a sphere of radius "a" centered on the origin by crafting a triple integral and solving for it using cylindrical coordinates. Homework Equations x^{2}+y^{2}+z^{2}=a^{2} : Equation for a sphere of radius "a" centered on...

Hey all, If anyone has some hints on how to do this one it would be much appreciated: Find the volume of the region given by x2 + y2 ≤ a2, 0 ≤ z ≤ x. So I've gone to cylindrical polars, and threw in the Jacobian, r. If I integrate with my bounds being: 0 to a, 0 to pi, 0 to...

Homework Statement Transform the equation from cartesians coordinates to spherical coordinates. Homework Equations \int_\infty\int_\infty\int_\infty exp\left\{ \frac{-\left| \vec{x'}-\vec{x}_0 \right|^2}{2 \sigma} \right\} \frac{\left( \vec{x} - \vec{x'} \right)}{\left| \vec{x} -...

Homework Statement Find the volume of the solid inside the sphere x^2 + y^2 + z^2 = 4 and over the paraboloid 3z = x^2 + y^2 The Attempt at a Solution This should be easy to calculate using polar coordinates. The limits for z is [r^2/2, sqrt(4-r^2)] and for tetha: [0, 2*pi], but how do...

[b]1. "Find the mass of part of the solid sphere x^2 + y^2 + z^ 2 ≤ 25 in the 1st octant x ≥ 0, y ≥ 0, z ≥ 0 where mass density is f (x, y, z ) = (x^2 + y^2 + z^2 )^3/2 ." [b]3. These problems are really stumping me! I need somebody to work it out/explain it to me! What will the limits...

Homework Statement [FONT="Courier New"] Find the volume of the tetrahedron in the first octant bounded by the coordinate planes and the plane passing through (1,0,0), (0,2,0) and (0,0,3) Homework Equations [FONT="Courier New"]V=∫∫∫dV [FONT="Courier New"]...D The Attempt at a...

hey! i need some help with a triple integration problem using spherical coordinates. it's the volume of a small part of a sphere. rho from 5 to 6, phi from pi/6 to pi/4 and theta from pi/4 to pi/3. i got an answer of (-91/72) x pi x (sqrt(2)-sqrt(3))...am i right? Thanks!

alo, i ve done single variable integration at school but i m trying to understand some vector analysis and going through books like Schey's Div, Grad Curl and all that as well as Schaum s vector analysis and to be able to understand i need to know exactly what a double integral is...i ve...

I have a group of problems that deals with the equations: f(x,y)= x^2+y^2 g(x,y)=20-(x-4)^2-(y+2)^2 Can someone help find the triple integral to find the volume.