# 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.

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1. ### Using Multiple integrals to compute expected value

I want to know how did author derive the red underlined term in the following Example?
2. ### Evaluation of integral having trigonometric functions

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...
3. ### I Is this integral using change of variables technique correct?

Is the answer given above correct?
4. ### I Which of the two answers is correct?

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?
5. ### Change of variables in multiple integrals

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 =...
6. ### HP 50g calculator's answer is correct or author's answer is correct?

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...
7. ### MHB Triple Integrals: Finding Volume of Solid S Bounded by Planes

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 ?
8. ### I Looking for an expert on integrals

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...
9. ### MHB Shortcuts for Solving Multiple Integrals: Is There a Faster Way?

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...
11. ### I Optimization of multiple integrals

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...
12. ### I Changing variables in multiple integrals

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...
13. ### I Multiple integral Jacobian confusion

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'}...
14. ### What is the volume inside an ellipsoid between two intersecting planes?

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...
15. ### I Change of order in double integrals

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...
16. M

### How to find the area of a triangular region using Green's Theorem

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...
17. ### MHB Change of Variables in Multiple Integrals

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...
18. ### I What other math courses involve multiple integrals?

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?
19. ### MHB Possible title: Troubleshooting File Attachments on Online Forums

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...
20. ### MHB Double & Triple Integrals: Same Solution?

when taking double or triple integrals, do you get the same solution no matter which variable you integrate with respect to first?
21. ### Integrals and Convolution: How to Group Multiple Functions Together?

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)...
22. ### Disk, Washer, Shell Multiple Integrals

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...
23. ### Multiple integrals: Find the volume bounded by the following surfaces

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...
24. ### Change of Variable in multiple Integrals

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)...
25. ### Find the exact volume of a bounded surface. Multiple Integrals.

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...
26. ### MHB Finding volumes by multiple integrals

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...
27. ### Multiple integrals for finding volume

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...
28. ### Converting cartesian to polar coordinates in multiple integrals

Homework Statement Do you see how y gets converted to csc? I don't get that. I would y would be converted to sin in polar coordinates.
29. ### Multiple integrals in polar form

Homework Statement do you see how the integral of r is .5? I don't get how that follows?
30. ### Solving Multiple Integrals - Understanding Triangle Area Calculation

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.
31. ### Understanding the meaning of multiple integrals

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...
32. ### Multiple Integrals for Functions Unbounded at Isolated Points

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...
33. ### Change of Variables in Multiple Integrals

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...
34. ### Mathematica Mathematica®: performing a varying number of multiple integrals

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...
35. ### Mathematica Non-integrable multiple integrals for Mathematica

“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...
36. ### Change of Variables multiple integrals

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...
37. ### Double Integral of (x-y)^2 (sin(x+y))^2 over a Square

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.
38. ### Finding the Volume of a Bounded Region in 3D Space using Multiple Integrals.

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
39. ### Multiple integrals applications of integration

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...
40. ### Calculating Multiple Integrals on Mathematica (Electric fields)

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...
41. ### Multiple Integrals Homework: Mass of Gold in Ore

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...
42. ### Help with Multiple Integrals Problems

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...
43. ### Change of variables in multiple integrals. What am I doing wrong?

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...
44. ### Change of Variables of f(x+y) (In Multiple Integrals)

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...
45. ### Evaluate Multiple Integrals with Polar Coordinates

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...
46. ### Change of variable in multiple integrals

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...
47. ### Changing the variable in multiple integrals

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...
48. ### Finding mass using multiple integrals

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...
49. ### Multiple Integrals with parallelogram

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...
50. ### Multiple Integrals: What They Do & How to Understand Them

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...