What is Triple integrals: Definition and 81 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.
If we solve the L.H.S. of this equation, we get ## \frac{(b-a)^3}{6}## and if we solve R.H.S. of this equation, we get ##-\frac{2b^3-3ba^2 +a^3}{6}##
So, how can we say, this equation is valid?
By the way, how can we use the hint given by the author here?
where the region of integration is the cube [0,1]x[0,1]x[0,1]
my question is where can we use the polar coordinate? is it only usable if the region of integration looks like a circle regardless of the function inside the integral? (if yes it means that using this kind of transformation is wrong...
ok this is a snip from stewards v8 15.6 ex
hopefully to do all 3 here
$\displaystyle\int_0^1\int_{0}^{1}\int_{0}^{\sqrt{1-x^2}}\dfrac{z}{y+1} \,dxdzdy$
so going from the center out but there is no x in the integrand
$\displaystyle\int_0^{\sqrt{1 - x^2}} \dfrac{z}{y + 1}dx =\dfrac{ \sqrt{1 -...
15.6.4 Evaluate the iterated integral
$$\int_0^1\int_y^{2y}\int_0^{x+y}
6xy\, dy\, dx\, dz$$OK this is an even problem # so no book answer
but already ? by the xy
Homework Statement
Let ##U_1, U_2, U_3## be independent uniform on ##[0,1]##.
a) Find the joint density function of ##U_{(1)}, U_{(2)}, U_{(3)}##.
b) The locations of three gas stations are independently and randomly placed along a mile of highway. What is the probability that no two gas...
$\textsf{The region in the first octant bounded by the coordinate planes and the surface }$
$$z=4-x^2-y$$
$\textit{From the given equation we get}$
\begin{align*}\displaystyle
&0 \le z \le 4-x^2-y\\
&0 \le y \le 4-x^2\\
&0 \le x \le z
\end{align*}...
Suppose $\displaystyle f = e^{(x^2+y^2+z^2)^{3/2}}$. We want to find the integral of $f$ in the region $R = \left\{x \ge 0, y \ge 0, z \ge 0, x^2+y^2+z^2 \le 1\right\}$.
Could someone tell me how we quickly determine that $R$ can be written as: $R = \left\{\theta \in [0, \pi/2], \phi \in [0...
Homework Statement
The first part of the question was to describe E the region within the sphere ##x^2 + y^2 + z^2 = 16## and above the paraboloid ##z=\frac{1}{6} (x^2+y^2)## using the three different coordinate systems.
For cartesian, I found ##4* \int_{0}^{\sqrt{12}} \int_{0}^{12-x^2}...
I recently came across a problem in Irodov which dealt with the gravitational field strength of a sphere. Took some time to get my head around it and figure how to frame a triple integral, but it felt good at the end. Am I going to start seeing triple integrals in the freshman year tho? If so...
Homework Statement
$$f(x,y,z)=y$$ ; W is the region bounded by the plane ##x+y+z=2##, the cylinder ##x^2 +z^2=1##, and ##y=0##.
Homework EquationsThe Attempt at a Solution
Since there is a plane of ##y=0##, I decided that my inner integral will be ##y=0## and ##y=2-x-z##. But after this I have...
Homework Statement
Let ##T \subset R^3## be a set delimited by the coordinate planes and the surfaces ##y = \sqrt{x}## and ##z = 1-y## in the first octant.
Write the intgeral
\iiint_T f(x,y,z)dV
as iterated integrals in at least 3 different ways.
Homework Equations
\iiint_T f(x,y,z)dV =...
Homework Statement
I will just post an image of the problem
and here's the link if the above is too small: http://i.imgur.com/JB6FEog.png?1Homework EquationsThe Attempt at a Solution
I've been playing with it, but I can't figure out a good way to "grip" this problem.
I can see some things...
Are these correct?
Thanks in advance!
1.) Set up the triple integral for ##f(x,y,z) = xy + 2xz## on the region ##0 ≤ x ≤4, 0 ≤ y ≤ 2## and ##0 ≤ x ≤ 3xy + 1##.
##\displaystyle \int_0^4 \int_0^2 \int_0^{3xy+1} 2y +2xz\ dz\ dy\ dx##
\text{2.) Set up the triple integral in cylindrical...
Homework Statement
My question is this: When finding center of mass, can you do so using spherical/cylindrical coordinates, or must you put it in cartesian coordinates?
If you can use spherical/cylindrical coordinates, how do you set up the triple integrals ?
Thank you.
Homework...
Homework Statement
a) sketch the region in the first octant bounded by the elliptic cylinder 2x^2+y^2=1 and the plane y+z=1.
b) find the volume of this solid by triple integration.
Homework EquationsThe Attempt at a Solution
I have already sketched the elliptic cylinder and the plane. my...
Hi all,
I'm not sure how to get the boundaries in terms of both the spherical and cylindrical coordinates for this question.
Here are the boundaries we were given in the solution.
How was \frac{\pi}{4} for φ and \frac{1}{\sqrt{2}} for r obtained?
Thanks!
Homework Statement
Assume that f(x,y,z) is a continuous function. Let U be the region inside the cone z=√x^2+y^2 for 2≤x≤7. Set up the intregal ∫f(x,y,z)dV over U using cartesian, spherical, and cylindrical coordinates.
Homework Equations
CYLINDRICAL COORDINATES
x=rcosθ
y=rsinθ
z=z...
I hope this makes my question clear...
suppose we have a triple integral of dzdydx for [0<x<1 , sqt(x)<y<1 , 0<z<1-y] and from the sketch we can see that 0<y<1 and 0<z<1...
my question is this, if we change the integration to dzdxdy we get [0<x<y^2 , 0<y<1 , 0<z<1-y], is that the only way? or...
Homework Statement
Calculate the volume of the body that is bounded by the planes:
x+y-z = 0
y-z = 0
y+z = 0
x+y+z = 2
Homework Equations
The Attempt at a Solution
I made a variable substitution
u = y+z
v = y-z
w = x
which gave me the new boundaries
u+w = 2...
Homework Statement
Evaluate ∫∫∫\sqrt{x^{2} + y^{2}} dA where R is the region bounded by the paraboloid y=x^2+z^2 and the plane y=4
Homework Equations
I believe this is a problem where cylindrical coordinates would be useful
0 ≤ z ≤ \sqrt{4-x^2}
0 ≤ r ≤ 2 ( I think this is wrong).
0 ≤ θ ≤...
Homework Statement
Find the Volume of the solid eclose by y=x^{2}+z^{2} and y=8-x^{2}-z^{2}
The Attempt at a Solution
Well know they're both elliptic paraboloids except one is flipped on the xz-plane and moved up 8 units. Knowing this, i equated the two equations and got...
Homework Statement
I want to know if I've gone about setting up these integrals in these questions properly before I evaluate them.
(i). Find the mass of the cylinder S: 0 ≤ z ≤ h, x^2 + y^2 ≤ a^2 if the density at the point (x,y,z) is δ = 5z^4 + 6(x^2 - y^2)^2.
(ii). Evaluate the...
Homework Statement
Write out the triple integral for the volume of the solid shown in all six possible orders. Evaluate at least 2 of these integrals.
Homework Equations
I attached a picture of the figure. The front : x/2+z/5=1...
Homework Statement
Despite the fact that this started as an extended AP Physics C problem, I turned it into a calc problem because I (sort of) can. If it needs to be moved please do so.
There is a hollow solid sphere with inner radius b, outer radius a, and mass M. A particle of mass m...
Homework Statement
Set up triple integrals for the integral of f(x,y,z)=6+4y over the region in the first octant that is bounded by the cone z=(x^2+y^2), the cylinder x^2+y^2=1 and the coordinate planes in rectangular, cylindrical, and spherical coordinates.
Homework Equations...
Homework Statement
My first problem is with 2ia) and 2ib), I got the correct answer, although not happy with my understanding of it.
http://img826.imageshack.us/img826/1038/443pr.jpg
The Attempt at a Solution
(2ia and 2ib)
The region that's of concern is the upper part between y = x^2 and...
Homework Statement
Find the volume of the solid that lies above the cone z = root(x2 + y2) and below the sphere x2 + y2 + x2 = z.
Homework Equations
x2 + y2 + x2 = ρ2
The Attempt at a Solution
The main issue I have with this question is finding what the boundary of integration is for ρ. I...
Homework Statement
Find the mass of a solid of constant density that is bounded by the parabolic cylinder x=y2 and the planes x=z, z=0, and x=1.
The Attempt at a Solution
https://dl.dropbox.com/u/64325990/Photobook/Photo%202012-06-07%202%2033%2024%20PM.jpg
I first drew some diagrams to...
Homework Statement
Use a triple integral to find the volume of the region. Below x+2y+2z=4, above z=2x, in the first octant.
Homework Equations
V=∫∫∫dV=∫∫∫dxdydz
The Attempt at a Solution
I have no clue where to begin as to finding those darn limits to integrate with. I'm sure...
Homework Statement
Find the mass m of the pyramid with base in the plane z = 9 and sides formed by the three planes y = 0 and y - x = 5 and 6x + y + z = 28, if the density of the solid is given by δ(x,y,z) = y.
Homework Equations
The Attempt at a Solution
This problem is driving...
Homework Statement
Find the volume of the solid enclosed between the cylinder x2+y2=9 and planes z=1 and x+z=5Homework Equations
V=∫∫∫dz dy dzThe Attempt at a Solution
The problem I have here is setting the integration limits. I first tried using:
z from 1 to 5-x
y from √(9-x2) to -3
x from -3...
The solid enclosed by the cylinder x^2 + y^2 = 9 and the planes y + z = 5 and z=1.
The biggest part for me (usually) is just being able to find my limits of integration for these problems (any suggestions about that would also be greatly appreciated). I think I found the correct limits for...
Homework Statement
Evaluate the integral, where E is the solid in the first octant that lies beneath the paraboloid z = 9 - x2 - y2.
∫∫∫(2(x^3+xy^2))dV
Homework Equations
x=rcosθ
y=rsinθ
x^2+y^2=r^2
The Attempt at a Solution
θ=0 to 2π, r=0 to 3, z=0 to (9-r^2)...
Homework Statement
A cylindrical coffee cup (8 cm in diameter and 10 cm tall) is filled to the brim
with coffee. Neglecting the weight of the cup, determine the torque at the handle
(2 cm from edge of cup 5 cm up from bottom of cup).
The easy way would be to just use the center of mass of the...
Hey,
I was just going through my vector calc textbook for this year and everything was going well until I reached double and triple integrals. My problem is the whole symmetry thing; when does (forgive me, I can't figure out the symbols) the integral from a to b become twice the integral from...
1. Find the volume, using triple integrals, of the region in the first octant beneath the plane 2x+3y+2z = 6
2. http://tutorial.math.lamar.edu/Classes/CalcIII/TripleIntegrals.aspx
SOLUTION:
1. Assume X and Y are 0. Solve for Z: 2(0)+3(0)+2z=6 => z=3 (0,0,3)
2. Assume X...
Write a triple integral to represent the volume of the solid
The wedge in the first octant and from the cylinder y^2 + z^2 <= 1 by the planes
y=x, x=0, z=0
First..
i find the range for z..; 0 <=z<= sqrt(1- y^2)
then...
i find the range for y..; let z =0
0<=y<=1
next, if i...
Homework Statement
The question is to use MATLAB to evaluate a triple integral in spherical coordinates to find the mass density of the solid inside the cone z = (3x^2 + 3y^2)^.5 and below z = 5 where the mass density at (x,y,z) is equal to the z coordinate of the point.
Homework...
Homework Statement
Use a triple integral to calculate the volume of the solid enclosed by the sphere
x^2 + y^2 + z^2=4a^2 and the planes z=0 and z=a
Homework Equations
Transform to spherical coordinates (including the Jacobian)
The Attempt at a Solution
I'm stuck, as the radius...
Homework Statement
\int\int\int^{}_{B} ye^(-xy) dV where B is the box determined by 0 \leq x \leq 4, 0 \leq y \leq 1, 0 \leq z \leq 5.Homework Equations
The Attempt at a Solution
\int^{4}_{0}\int^{1}_{0}\int^{5}_{0} ye^(-xy) dzdydx
Integrating the first time I get
zye-xy
Plugging in 5 and 0 I...
Homework Statement
Find the centroid x,y,z of the region R cut out of the region 0<=z<=5sqrt(x2+y2) by the cylinder x2+y2=2x.
Homework Equations
x2+y2 = r2
x= rcosθ
y= rsinθ
The Attempt at a Solution
Centroid x being Mx/m I'm guessing
I've been working on this problem...
Homework Statement
Solve for the volume above the xy-plane and below the paraboloid z=1-x2/a2-y2/b2
I have gotten an answer that is close to the correct one, but I can't figure out where I am wrong.
Homework Equations
Solution: Volume is = ab\pi/2
The Attempt at a Solution...
Homework Statement
Volume between Y=1-X and Y = Z^2 -1
The Attempt at a Solution
http://img254.imageshack.us/img254/1743/42932830.jpg
Uploaded with ImageShack.us
Sorry, not a good drawer.
0 < y < 1
0< x < SqRoot: 1-y
-1 < y < 0
0 < Z < Sqroot: 1 + y
I'm not even sure...
Homework Statement
The volume of a solid of revolution using the shell method is \int_{a}^{b} 2\pi x f(x) dx. Prove that finding volumes by using triple integrals gives the same result. (Use cylindrical coordinates with the roles of y and z changed).
Homework Equations
dV = r dr d\theta...
Homework Statement
Rewrite this integral the other five ways
\int_{x=0}^{1}\int_{z=0}^{1-x^2}\int_{y= 0}^{1-x} dydzdx
Homework Equations
Must be in rectangular coordinates
The Attempt at a Solution
1.)\int_{z=0}^{1}\int_{x=0}^{\sqrt{1-z}}\int_{y= 0}^{1-x} dydxdz...
so i have to find the moment of inertia of a solid cone given by the equations z = ar and z = b by using a triple integral. The density of the cone is assumed to be 1. so the integral looks like ∫ ∫ ∫ r^2 dV. so first i did it with dV = rdrdθdz with limits r (from 0 to z/a), θ (from 0 to 2pi)...
Homework Statement
Here is a solved problem:
[PLAIN]http://img3.imageshack.us/img3/6948/97765276.gif
In part (e), they formulated the triple integral using the limits of integration they found by sketching the region. Is there a way we can find the limits of integration without...