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.
Homework Statement
Hi, I've been trying this for days now and I really can't get it, so would appreciate some help please!
Find the volume of the finite region between the two surfaces z=x^2 + 4y^2 and z=2x + 8y + 4
Homework Equations
The Attempt at a Solution
I tried...
I find this to be a very tough problem:
1) Determine whether the improper integral I
∞ ∞
∫ ∫ [sin (x2 + y2) / ln(x2 + y2)] dxdy
2 2
converges or diverges.
All I can think of and try is by changing it to polar coordinates:
I=A+B where
A=
pi/4--- ∞
∫ ----- ∫ [sin (r^2) /...
I am having some trouble with the following 2 questions on improper multiple integrals. I hope that someone can help me out!
1) Determine whether
I=∫∫ cos(sqrt(x2+y2)) / (x2+y2) converges or diverges.
x,y>1
Solution:
Let R=[0,1]x[0,1]
B(0,1)=ball of radius 1 centered at origin...
Q1) Let f(x,y)=3, if x E Q
f(x,y)=2y, if y E QC
Show that
1 3
∫ ∫ f(x,y)dydx exists
0 0
but the function f is not (Riemann) integrable over the rectangle [0,1]x[0,3]
I proved that the iterated integral exists and equal 9, but I am completely stuck with the second part (i.e. to prove...
Q1: Let S be the region in the first quadrant bounded by the curves xy=1, xy=3, x2 - y2 = 1, and x2 - y2 = 4. Compute
∫∫(x2 + y2)dA.
S
(Hint: Let G(x,y)=(xy, x2 - y2). What is |det DG|?)
Solution:
http://www.geocities.com/asdfasdf23135/advcal19.JPG
I don't understand the third and...
Q1: Suppose B=[0,1]x[0,2]x[0,3]x[0,4] in R4, and that C=[0,1]x[0,1]x[0,1]x[0,1]. Given that
∫ ∫ ∫ ∫f(x)=d4x=(2pi)4
B
What is the value of
∫ ∫ ∫ ∫ f(x1,2x2,3x3,4x4) d4x?
C
Solution:
Define x=G(u)=(u1,u2/2,u3/3,u4/4)
∫ ∫ ∫ ∫ f(x1,2x2,3x3,4x4) d4x
C
by change of variables theorem...
1) Find the volume of T bounded below by the cone z=sqrt(x2+y2) and above by the sphere x2+y2+z2=1.
Solution:
Volume =
∫∫∫ 1 dV =
T
b d f
∫ ∫ ∫ r (d theta)dzdr (change of variables to cylindrical coordinates)
a c e
where
a=0
b=1/sqrt2 <---I am having a lot of trouble...
Homework Statement
(Q) Compute ∬_R▒(y-2x^2 )dA where R is a region bounded by the square |x| + |y| = 1.
Homework Equations
The Attempt at a Solution
The absolute functions are throwing me all over the place and I am not even able to begin:cry:
Find the volume of that portion of a unit sphere for which 0<theta<alpha, where theta is one of the spherical coordinates
So i know the equation is z^2 = x^2 + y^2 , but what is the meaning of 0<theta<alpha?
where do i start? I know i must convert to polar coordinates.
z^2 = r^2
i was looking through a book and came across a double integral that was split into the product of two single integrals.
it was int (x^n)(y^n ) dxdy split into (int x^n dx)(int y^n dy)
i just finished a course in multivariable calculus(it was by no means thorough), and i didn't know that...
Hi,
I need some help on these problems. I'm not sure what to do.
1 Find the area of the plane with vector equation r(u, v) =< 1+v, u-2v, 3-5u+v> that is given by 0<u<1, 0<v<1.
So far, I took the partial derivatives with respect to u and v. I don't know if I was supposed to or not and I'm...
Surface Area (help me to prove something:)
I was studying a bit about multiple integrals and found this theorem:
If we have function z=f(x,y) which is defined over the region R, surface S over the region is
S=\iint_R\sqrt{1+\left(\frac{\partial f}{\partial x}\right)^2+\left(\frac{\partial...
Homework Statement
Hello, I was wondering if someone could help me with the following. Supposed I am asked to find the volume bounded by the cylinders x^2+y^2=1 and the planes y = z, x = 0, z = 0 in the first octant.
Homework Equations
So this is what I tried to do. The boundaries...
How do you make the limits in a triple integral look okay? I need to write something like:
\iiint_{x \geq 3, y \geq 4, z \geq 5, 2z - x \geq 5} f(x, y, z)\, dx\,dy\,dz
but it looks kind of silly right now.
How do I go about showing the region of integration represented by a repeated integral? (Just for 2 dimensional functions)
All the diagrams I've seen show the area under the graph, but with like 2 bars across it. I don't understand what all this represents and my poor quality notes are of...
I am currently having trouble solving this problem "Find the volume of a right, circular cone of radius r and height h" a) as a single integral. b) as a double integral. c) as a triple integral.
My difficulty lies in setting up the integrals. I usually have trouble with problems that do not...
Hi, I posted a question some time ago and the suggestion was to use some form of the product rule but I still can't figure out what to do.
Q. Let f(x,y,z) and g(x,y,z) be C^2 scalar functions. Let D be an elementary region in space and \partial D be the closed surface that bounds D. Prove...
if i have a hemisphere of radius 4, is it possible using multiple integration for me to find the radius of a cylinder that sits inside the hemisphere such that the vol inside the hemisphere and outside the cylinder is a 1/12 of the vol of the hemisphere
anyone that can help me on this-i...