What is Triple integration: Definition and 36 Discussions
In mathematics (specifically multivariable calculus), a multiple integral is a definite integral of a function of several real variables, for instance, f(x, y) or f(x, y, z). Integrals of a function of two variables over a region in
R
2
{\displaystyle \mathbb {R} ^{2}}
(the real-number plane) are called double integrals, and integrals of a function of three variables over a region in
R
3
{\displaystyle \mathbb {R} ^{3}}
(real-number 3D space) are called triple integrals. For multiple integrals of a single-variable function, see the Cauchy formula for repeated integration.
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...
Volume of Frustum Using Triple Integral [Solved]
Homework Statement
Edit: I've solved the issue! My limits of r were wrong. Instead of this:
V=\int_{z=0}^{\frac{h}{2}}\int_{r=0}^{\frac{R}{h}z+R}\int_{\theta=0}^{2\pi}r\ d\theta\ dr\ dzIt should have been this...
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
∫∫∫...
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
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
Evaluate the following integral by changing to cylindrical coordinates:
I displayed the question and my attempt in the document attached.
Homework Equations
The Attempt at a Solution
The attempt is in the document attached. Please help me to check whether I...
Homework Statement
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
V=∫∫∫dV
...D
The Attempt at a Solution
I set up the problem as so:
1 -2x+2...-3x+3
∫...
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.