Multiple integrals Definition and 68 Threads

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[/color] Solution: Define x=G(u)=(u1,u2/2,u3/3,u4/4) ∫ ∫ ∫ ∫ f(x1,2x2,3x3,4x4) d4x C by change of variables...

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[/color] <---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...