What is Multiple integrals: Definition and 68 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.
R is the triangle which area is enclosed by the line x=2, y=0 and y=x.
Let us try the substitution ##u = \frac{x+y}{2}, v=\frac{x-y}{2}, \rightarrow x=2u-y , y= x-2v \rightarrow x= 2u-x + 2v \therefore x= u +v##
## y=x-2v \rightarrow y=2u-y-2v, \therefore y=u- v## The sketch of triangle is as...
Solution 1:
The answer is ## \frac{2b^5\pi}{5} \times \left(1 -\frac{a}{\sqrt{1+a^2}}\right)##
Solution 2:
I want to decide which answer is correct? Would you help me in this task?
Summary: Find the volume V of the solid inside both ## x^2 + y^2 + z^2 =4## and ## x^2 +y^2 =1##
My attempt to answer this question: given ## x^2 + y^2 +z^2 =4; x^2 + y^2 =1 \therefore z^2 =3 \Rightarrow z=\sqrt{3}##
## \displaystyle\iiint\limits_R 1dV =...
Summary: Evaluate ##\displaystyle\iint\limits_R e^{\frac{x-y}{x+y}} dA ## where ##R {(x,y): x \geq 0, y \geq 0, x+y \leq 1}##
Author has given the answer to this question as ## \frac{e^2 -1}{4e} =0.587600596824 ## But hp 50g pc emulator gave the answer after more than 11 minutes of time...
Find the volume V of the solid S bounded by the three coordinate planes, bounded above by the plane x + y + z = 2, and bounded below by the z = x + y.
How to answer this question using triple integrals? How to draw sketch of this problem here ?
I’ve written an insight article on what I think is original material (at least I’ve not seen it in my reading nor google):
A Novel Technique of Calculating Unit Hypercube Integrals
I am looking first for someone that can follow my work, I’ve had some mathematicians look over it but none whose...
I'm having a problem solving this, My approach is solving $x_1$ as a variable and rest as constants first and then going on further. But it is getting too lengthy. Is there any short method?
The following 3 pages are extract from the book: "CALCULUS VOL II" by Tom M. Apostol
My interpretation of these 3 pages is worked out in the attached PDF file. Entirely done in propositional logic language.
Can anyone point out the mistakes or incorrect logical steps (if any) in the attached...
The Euler Lagrange equation finds functions ##x_i(t)## which optimizes the definite integral ##\int L(x_i(t),\dot x_i(t))dt##
Is there any extensions of this to multiple integrals? How do we optimize ##\int \int \int L(x(t,u,v),\dot x(t,u,v))dtdudv## ?
In particular I was curious to try to...
Suppose we have a region R in the x-y plane and divide the region into small rectangles of area dxdy. If the integrand or the limits of integration were to be simplified with the introduction of new variables u and v instead of x and y, how can I supply the area element in the u-v system in the...
Consider a continuous charge distribution in volume ##V'##. Draw a closed surface ##S## inside the volume ##V'##.
___________________________________________________________________________
Consider the following multiple integral:
##\displaystyle B= \iint_S \Biggl( \iiint_{V'}...
Homework Statement
Find the volume between the planes ##y=0## and ##y=x## and inside the ellipsoid ##\frac{x^2}{a^2} + \frac{y^2}{b^2} + \frac{z^2}{c^2} = 1##The Attempt at a Solution
I understand we can approach this problem under the change of variables:
$$x=au; y= bv; z=cw$$
Thus we get...
In the question given below, can we change the order of integral so that y can be the independent variable and x be the dependent one?The cylinder x^2 + z^2 = 1 is cut by the planes y=0,z=0 and x=y.Find the volume of the region in the first octant.This may look like a homework question but it's...
Homework Statement
You have inherited a tract of land whose boundary is described as follows. ”From the oak tree in front of the house, go 1000 yards NE, then 1200 yards NW, then 800 yards S, and then back to the oak tree.
Homework Equations
Line integral of Pdx + Qdy = Double integral of...
Let S S = double integrals
S S x^2 dA; where R is the region bounded by the ellipse
9x^2 + 4y^2 = 36.
The given transformation is x = 2u, y = 3v
I decided to change the given ellipse to a circle centered at the origin.
9x^2 + 4y^2 = 36
I divided across by 36.
x^2/4 + y^2/9 = 1
I replaced...
Didn't know where to put this question. but just wanted to ask quickly. All i know is that in Calculus 3, you use double, triple or multiple integrals for 3D models. But what other higher level math course uses or involves multiple integrals?
a) The two surfaces defined by xz^2 + 2yz +xy^2 = 4, 2x - 7y - 3z = 2 intersect in a curve S. Find the tangent line and normal plane to S at the point (3,1,-1).
b) Sketch the graphs of y = x^m and y = x^n where m and n are...
Hi,
I shall show the following:
(f*g) \star (f*g) = (f\star f)*(g\star g)
where * denotes convolution and \star cross-correlation. Writing this in terms of integral & regrouping:
\int_{\phi} \left(\int_{\tau_1} f(t - \tau_1) g(\tau_1) d\tau_1\right) \cdot \left(\int_{\tau_2} f(\tau_2)...
Homework Statement
Determine how many integrals are required for disk, washer, and shell method.Homework Equations
x=3y^2 - 2 and x=y^2 from (-2,0) to (1,1) about x-axis.The Attempt at a Solution
Since there are no breaks or abnormalities in the graph it appears that 1 integral will solve for...
Homework Statement
Find the volume bounded by the following surfaces:
z = 0 (plane)
x = 0 (plane)
y = 2x (plane)
y = 14 (plane)
z = 10x^2 + 4y^2 (paraboloid)
Homework Equations
The above.The Attempt at a Solution
I think it has something to do with triple integrals? But...
Homework Statement
Let D be the triangular region in the xy-plane with the vertices (1, 2), (3, 6), and (7, 4).
Consider the transformation T : x = 3u − 2v, y = u + v.
(a) Find the vertices of the triangle in the uv-plane whose image under the transformation T is the triangle D.
(b)...
1. Ok, so the question is.. Find the exact volume of the solid bounded above by the surface z=e^{-x^2-y^2}, below by the xy-plane, and on the side by x^2+y^2=1.
2. Alright. So, I know that I can use a double integral to find the volume, and switching to polar coordinates would be simpler...
How do I solve this? How do I determine the range? Ill they be triple integrals?Please explain to me.
Find the volumes in R3.
1. Find the volume U that is bounded by the cylinder surface x^2+y^2=1 and the plane
surfaces z=2, x+z=1.
2. Find the volume W that is bounded by the cylindrical...
How do I solve this? How do I determine the range? Ill they be triple integrals?Please explain to me.
Find the volumes in R3.
1. Find the volume U that is bounded by the cylinder surface x^2+y^2=1 and the plane
surfaces z=2, x+z=1.
2. Find the volume W that is bounded by the cylindrical...
Homework Statement
The Attempt at a Solution
I understand the steps, although it took quite a while, but what I don't understand is that a triangle with base 2 and height 2, it's area is 2. With two triangles of that size the area should be 4. The books says the area is 8.
Homework Statement
I am currently taking calc III and we have starting getting into double and triple integrals. I was wondering what you are actually doing when you take a double or triple integral? And what the difference is. I understand that you find area with a single integral and find...
In a recent homework assignment, I was asked to prodive a definition for ∫f(x) in the Region D, provided there was a discontinuity somewhere in the region. To define the integral, we merely removed a sphere centered on the discontinuity of radius δ>0 and found the limit of the integral as δ→0...
The problem is:
R is the parallelogram bounded by the lines x+y=2, x+y=4, 2x-y=1, and 2x-y=4. Use the transformation u=x+y and v=2x-y to find the area of R.
I am not sure how to complete this problem. My first issue is that I don't know how to convert the transformation functions into...
Hello everyone.
In Mathematica® I want to numerically integrate a function of k variables (k varies) with respect to all of them. Does anyone of you know a way to do that? I tried the following simplified example.
k = 5;
int[x_] := x[[1]] + x[[2]] + x[[3]] + x[[4]] + x[[5]] ; (* My...
“Non-integrable” multiple integrals for Mathematica
Dear all,
I have been trying to crack one problem in Mathematica, but I keep getting a wrong answer probably because I have something either fundamentally wrong analytically or code wise. OK, here is the problem.
Suppose we have to...
Homework Statement
Find the volume of the cone bounded below by z=2root(x2+y2) and above by x2 + y2 + z2 = 1
Homework Equations
The Attempt at a Solution
Ok I have the solution, I just don't understand how to get it!
So I know I have to change into spherical coordinates but...
Homework Statement
Evaluate double integral (x-y)^2 (sin (x+y))^2 dxdy taken over a square with successive vertices (pi,0), (2pi,pi), (pi,2pi), (0,pi)
Thank you.
vollume bounded by
x+y=4
y^2+4z^2=16
not sure how to set this up
also
I 6y+x dx + y+2x dy
along
(x-2)^2+(y-3)^2=4
also
vollume bounded above by
z=4-4(x^2+y^2)^2
below by
(x^2+y^2)^2-1
Homework Statement
Find the mass of the solid from problem 42 if the density is proportional to y
42. Find the volume between the planes z=2x+3y+6 and z=2x+7y+8 and over the triangular vector (0,0) (0,3) and (2,1)
The Attempt at a Solution
The reason I'm asking is because I'm not sure...
I'm trying to calculate several stuff about the electric field generated by 2 "plaques"
My main problem being that in mathematica it calculates fast for a point of the electric field generated in the Y component, but if i try to make the median of all the points in the plaque it takes way to...
Homework Statement
A hemispherical piece of ore of radius a contains flakes of gold. The flat base
of the ore is in the x, y plane and its curved surface is in the region z > 0,
where x, y, z are cartesian co-ordinates with origin at the centre of the base.
The gold density is kz kg m−3, where...
Hi,
I have some conceptual problems regarding multiple integrals,out of which some often make me do sums wrong. Please help me out!
1. If we triple integrate a function f(x,y,z) within appropiate limits (there are some sums of this kind in my book) are we integrating in...
the problem:
evaluate the following integral by making appropriate change of variables.
double integral, over region R, of xy dA
R is bounded by lines:
2x - y = 1
2x - y = -3
3x + y = 1
3x + y = -2
my attempt:
let 2x - y = u, and let 3x + y = v
then the new region in (u,v) coordinates...
Homework Statement
The problem is as follows: Let T be the triangle with vertices (0,1), (1,0), (0,0). Compute the integral \int\int\frac{sin^{2}(x+y)}{(x+y)} dxdy by making an appropriate change of variables. (Hint: check #24 Section 15.9)
Homework Equations
Problem 24 in 15.9 of...
Homework Statement
By changing to polar coordinates, evaluate:
\int\int e ^(-\sqrt{x^2 + y^2}) dx dy
Both integrals go from 0 --> infinity
Homework Equations
r = \sqrt{x^2 + y^2}
x = r cos\theta
y = r sin\theta
Using the Jacobian to switch to polar coord we get:
J = r...
Hello. Does someone has studied the Change of Variables Theorem for multiple integrals in Apostol's Mathematical Analysis? (First Edition:not Lebesgue but Riemann).
I hope that some of you has the same edition, because if not, it will be sort of dificult to make a legible copy of the...
Homework Statement
Evaluate
\int\int(x-y)^2sin^2(x+y)dxdy
taken over a square with successive vertices (pi,0), (2pi,pi), (pi,2pi), (0,pi).
Homework Equations
I = \int\int_{K} f(x,y)dxdy = \int\int_{K'} g(u,v)*J*dudv
where J is the Jacobian.
The Attempt at a Solution...
The density per unit area of a circular lamina of radius a varies as the cube of the distance from a single point on the edge. Find the mass of this lamina.
im guessing id have to do ρdxdydz, and maybe use polar coordinates but I am completely lost. I am used to the question giving me an...
Use the given transformation to evaluate the given integral, where R is the parallelogram with vertices (-2, 2), (2, -2), (4, 0), and (0, 4).
∫∫(2x+8y)dA; x=1/2(u+v) y=1/2(v-u)
I found the bounds of the parallelogram of -4≤u≤4 and 4≤v≤0
so i set the equation to be...
I'm just curious of what exactly multiple integrals are for example if you have
\int^{3}_{1}xdx
you get 3-1[i think - it's been a while] but what does the second integral or 3rd and so on do to the function, I've looked ahead in my solutions manual and i think i understand part of it :D but...