Double integral Definition and 558 Threads

Homework Statement Evaluate (using a substitution) \int\int_{B}x^{2}+2y dxdy where B=\{(x, y) | x^{2}+y^{2}≤4\} The Attempt at a Solution I attempted a solution using polar coordinates, so the integral becomes \int\int_{B_{ρθ}}(ρ^{2}cos^{2}(θ)+2ρsin(θ)) ρ dρdθ, and the integration...

Hi there. I think I have proved on little theorem on double integrals, showed below. Are my arguments 'correct' (I mean, rigorous enough)? Let f be a function of x, f(x), g be a function depending only on y, g(y), and last, let A be the set determined by a≤x≤b and c≤y≤d. By Fubini's theorem...

Homework Statement Integrate the following function f over the given region Ω: f(x,y) = xy; Ω bounded by y = 0, x = 2a, and x^2 = 4a The Attempt at a Solution The given answer is (a^4)/3 The answer I got was (a^4)...ignore the answer I put in the attachment at the end, slight...

Homework Statement ...Bounded by graphs of equations: z=xy, z=0, y=x, x=1 I don't know what z=xy is. The rest of boundaries are clear. I assume that when y=1 and x=1, z=1. But, is this a z=1 plane? Check my figure attached. Thank you. Homework Equations The Attempt at a Solution

Problem: Can anyone help me out with the following problem: I am given a uniformly continuous function : g:\mathbb{R}^{2}\rightarrow [0,\infty ) such that the following condition is satisfied: \sup_{r> 0}\iint_{{x^{2}+y^{2}\leq r^{2}}}g(x,y)dxdy< \infty The question is to prove that:\lim_{|...

Homework Statement Find the area of a square with each side measuring 1 using double integral and change of euclidean coordinates to polar coordinate. Homework Equations x=rcosθ y=rsin0 dA=dxdy=rdrdθ The Attempt at a Solution int(int(rdr)dθ)

Hi, I wrote a piece of MATLAB code to compute a double integral of the form: \int_{a}^{b}\int_{c}^{d}f(x,y)dxdy I went about it using the trapezium rule, so what I did was apply the rule to the x variable first to obtain: \int_{a}^{b}\int_{c}^{d}f(x,y)dxdy\approx\frac{\delta...

Suppose we have: ∫∫ cos(x/y) dydx where the first integral is of x and is 0→1, while the second is of y and is x→1. Could someone tell me how to get the first integration (i.e. of cos(x/y) w.r.t. y) done??

Suppose the question is: ∫∫exp(y)/y dydx Now the first integral is w.r.t. y and goes from 0 to 1. The second integral is that of x and goes form y to y^2. I've evaluated it multiple times and it comes out to be 2-e. ( I first integrate the integrand w.r.t. x then w.r.t. y as the other...

\int xy^3 dx+ x^5 dy, where C is the rectangle with vertices (0,0), (4,0), (4,2), and (0,2) F(bar)= <P,Q> <xy^3, x^5> derivative of P with respect to y= 3xy^2 derivative of Q with respect to x= 5x^4 Double \int (5x^4-3xy^2) dx dy with limits for x from (0,2) and y limits (0,4) I get 0 for...

Could somone tell me how is it that the double integral could be used for both calculating the area as well as the volume? And please explain that how does the triple integral, which is used to find the volume as well, fits in the picture and how is it different from the double integral we use...

Homework Statement Use a double integral to find the area of the region bounded by the curve r= 1+sin(theta)? Homework Equations The Attempt at a Solution I can't figure out what theta is intregrated from. I've tried from -(pi)/2 -> +(pi)/2 and that doesn't work. I've also tried...

Can the TdS by a system be calculated by taking the area, time double integral of the heat flux density? If so, is it possible that this double-integral would take on an opposite sign if inside there was a dominating, growing black hole, where, I would presume, heat flows inwards, and not...

Homework Statement Evaluate \int\int (2x+1)(x-y)dxdy where Ω is the region in the first quadrant of the x-y plane bounded by: y=x, y=x+2, y=2-x^2 and y=4-x^2 The attempt at a solution This is the graph of the intersecting curves and the shaded area is the region Ω...

Homework Statement I'm trying to determine the limits for a double integral over a symmetric trapezoid or equilateral triangle. I'm not trying to determine the area, and therefore using symmetry to simplify the integration is not an option. The limits for the integration over the y-axis are...

Homework Statement $$\left[\int\limits_0^{Inf} {\int\limits_0^{Inf} {e^{ - aX - bY} \cdot F(X + Y + c)} }\cdot X^d \cdot Y^e \cdot dX \cdot dY\right]$$ Homework Equations a,b,c are constants; d & e are non negative integers; X and Y are variables. F is a one to one function. Please simplify. The...

Homework Statement Find the volume of the solid of revolution of the area bounded by the curves about the given axis. y= x^2 - 2, y = 0, about y = -1, consider only the area above y=-1 Homework Equations The Attempt at a Solution So I drew out the problem and figured it would be...

Homework Statement Let f(x,y) = sin(∏*y^2). Let R be the triangular region on the x-y plane with the vertices at (0,0) (0,1) (.5,1). Consider the solid that is under z = f(x,y) and over the region R. Calculate the volume over that region using double integrals. Homework Equations The...

Hi all. Suppose that we want to compute the following indefinite integral: The correct solution by Mathematica: Now here is the (apparently) incorrect solution by using polar coordinates: \iint\frac{1}{\sqrt{x^2+y^2}}dxdy=\iint\frac{1}{r}rdrd\theta=(r+c_1)(\theta+c_2) If...

Homework Statement http://img10.imageshack.us/img10/3390/55486934.jpg Homework Equations This is what I was thinking: tan−1(∏x)−tan−1(x)=∫^{g(x)}_{f(x)}h(y)dy The Attempt at a Solution I don't really understand how to do this question

Hi every one! I would like to draw a double integral function in related to R and h parameters by below M-File but It does goes wrong! Is there anyone to correct it for me? thank you syms R h; a1 = 0; a2 = atan(R./(R+h)); r1 = h; r2 = sqrt(R.^2+(R+h).^2); integrand =...

Homework Statement Evaluate the integral (x^2+y^2)arctan(y/x) for 0<y<a and 0<x<(a^2-y^2)^0.5. Homework Equations The Attempt at a Solution I tried changing the order of integration to get the integral (x^2+y^2)arctan(y/x) for 0<x<a and 0<y<(a^2-x^2)^0.5. I noticed that this was a quarter of...

I am wondering is there any general analytic solutions to the following integrals First moment of area Integral(xdxdy) Integral(ydxdy) Second moment of area) Integral(x^2dxdy) Integral(y^2dxdy) Integral(xydxdy) Over the triangle (x1,y1) (x2,y2) (x3,y3) thanks Wei

Homework Statement The volume underneath the surface z= y/ (1+xy) and above the square {(x,y)| 2≤x≤3 , 3≤ y≤ 4} is:Homework Equations Please see attachment. The Attempt at a Solution Please see attachment for solution. My professor had provided us with 8 possible solutions (where only one...

Homework Statement http://s2.ipicture.ru/uploads/20120109/dT4m6rNG.jpg The attempt at a solution x=\frac{u}{1+v} and y=\frac{uv}{1+v} Transforming the integrand: \frac{x+y}{x^2}e^{x+y}=\frac{(1+v)^2 e^u}{u} dxdy=J.dudv J=\frac{v(1+v)^2 +1+uv}{(1+v)^3} The double integral becomes: \int\int...

Homework Statement http://s2.ipicture.ru/uploads/20120107/vVVkUT7f.jpg The attempt at a solution I plotted the graph x-y: http://s2.ipicture.ru/uploads/20120107/ja3V9aSV.jpg y=\frac{1}{2}(u+v) and x=\frac{1}{2}(u-v) So, after finding the Jacobian, the double integral becomes: \int\int...

Homework Statement find ∫∫ sin(x+y) dxdy in the domain D where D=(x,y) where y≥√x and y≤2x and y≥(1/x) and y≤2/x Homework Equations i The Attempt at a Solution took y as variable so i have two domains D1 where x is between 0.5 and 1 and y between 1/x and √x D2 where x is between 1...

Evaluate ∫∫R 5x2 + 2y2 where R is triangle (1,1) (2,0) (2,2) I see the lines bounding the triangle are y=x y=2-x and x=2, and have tried many attempts at setting up the correct limits. Would it be correct to split this into 2 triangles, or are the limits y=x∫y=2-x for y and 2∫1...

Homework Statement Find area of region bounded by x^2 + y^2 = 4above y=1 The attempt at a solution OK, so i drew the graph. http://s1.ipicture.ru/uploads/20111226/oR99IdxJ.jpg The red part of the graph is the area that i need to find. Put y = 1 in the equation of circle; x^2 = 3 x = -√3 and...

Homework Statement Evaluate the double integral: http://s1.ipicture.ru/uploads/20111224/K3MGcdS7.jpg The attempt at a solution Consider x as constant, first integrate w.r.t.y. I get \frac{xy^2}{2} After evaluating the integral with limits y=0 to y=y, i get the same exact answer. Then, i...

Homework Statement Find the mass of the plane region R in the first quadrant of the xy plane that is bounded by the hyperbolas xy=1, xy=2, x^2-y^2 = 3, x^2-y^2 = 5 where the density at the point x,y is \rho(x,y) = x^2 + y^2. Homework Equations The Attempt at a Solution The...

Homework Statement ∫0 to 2 ∫x/4 to 1/2 (sin (pi*y2)) dy dx Homework Equations The Attempt at a Solution I think I have to convert this to polar or do some sort of change of variable. Although in polar y = r sin θ, so then you would have sin of a sin??

Homework Statement Here is the question Homework Equations The Attempt at a Solution So converting to polars by x = r cos θ, y = r sin θ gives me r^5 in the inner integral but how do you convert the ranges? y goes from 0 to sqrt 1 - x^2, so that is just r, I could tell just by looking...

Homework Statement Evaluate the double integral by converting to polar coordinates. ∫∫ arctan y/x dA; R is the sector in the first quadrant between the circles 1/4= x^2+y^2 and x^2+y^2=1 and the lines y=x/√3 and y=x. Homework Equations arctan y/x= θ The Attempt at a Solution...

Homework Statement function inside is (x+y)^2 * sin(x^2 - y^2) R is the triangular region w/ vertices (0,0) , (0,2) , (1,1) x = (u+v)/2 y = (v-u)/2 What are the correct limits ?? The Attempt at a Solution Also, when plugging in x and y in the function, i ended up getting...

Homework Statement Evaluate: ∫1 to 4∫0 to y(2/(x^2+y^2))dxdy Homework Equations The Attempt at a Solution So I know you have to spilt it up and do the dx integral first: ∫0-y(2/(x^2+y^2))dx Now this is where I don't know if I'm doing it right, I moved the 2 outside the...

Homework Statement Evaluate. ∫∫D x2 dAxy, bounded by 5x2 + 4xy + y2 = 1 Homework Equations ∫∫D H(x,y) dAxy = ∫∫D H(u,v)\frac{\delta(x,y)}{\delta(u,v)}dAuv The Attempt at a Solution So I understand I'm supposed to find a change of variables to transform the ellipse into a circle...

I'm studying for my final and tutors/my professor isn't available over the weekend. Could someone please spend a little time to help me? My problem is stated as: Let R be the right half of the circle x2+(y-1)2=1. Use a double integral polar coordinates to find the area of the region R. I...

Homework Statement I just need to know if what i did is correct. The question is as follows: http://imgur.com/1RL7e Homework Equations The Attempt at a Solution What I did is as follows: I split this integral into two parts and solved. [integral from 0 to 1...

Homework Statement Evaluate the integral: integral D of y sin (x+y^2) dA where D = [0x2] x [0x2] U [1,3] x [1,3] Homework Equations The Attempt at a Solution So D is basically a square which simplifies to D = [1,2] x [1,2] since that is the portion of both rectangles that overlap. So then...

Homework Statement D = {x>0, x^2 < y < 10-x^2) compute integral (integral D of y^2 sqrt x) Homework Equations The Attempt at a Solution I'm having trouble figuring out the bounds of the integral. y goes from x^2 to 10-x^2 but I think I have to split this integral up...

How would you solve the following problem on Wolfram|Alpha (www.wolframalpha.com)? Problem If C is a circle or radius 1 with equation x^2+y^2=1, find \iint_{C} (x^2 + y^2) \cdot \mathrm{d}x \cdot \mathrm{d}y

Homework Statement 1 1/2 ∫ ∫ e-x2 dx dy 0 y/2 Homework Equations ***Graph equation*** The Attempt at a Solution I graphed the function and they made a horizontal strip. However, I can't seem to find the right function with a vertical strip, which is where I'm stuck.

I’m doing a lot of double integrals to find surface area problems, and I don’t think I’m setting them up quite right. For example, “Find the surface area of the portion of the sphere x^2 + y^2 + z^2 = 25 inside the cylinder x^2 + y^2 = 9.” I converted the sphere to a function of z: \sqrt{25...

Homework Statement Let R be the rectangle bounded by x - y = 0, x - y = 2, x + y = 0, and x + y = 3. Evaluate \int\int(x + y)ex2-y2dA R The Attempt at a Solution First I rewrote the boundaries so that I could graph them more easily. I got y = x, y = x - 2, y= -x, and y = -x + 3. I was going...

Homework Statement Evaluate the integral \int0 to 1\inty to 1\frac{1}{1+x^4}dxdy The Attempt at a Solution I managed to do the first one, from y to 1, using partial fractions and then some substitution, and I get a huge answer involving some logarithms and arctans that don't simplify...

The problem and my work is shown in the image below. However, I feel like I did something horrible wrong but I'm not sure where! I'm sorry if my handwriting is illegible. If you're having difficulties please leave a comment and I will not hesitate to type it out as a response. Any...

Homework Statement Evaluate over the x,y plane: ∫∫e^{-\sqrt{x^{2}+4y^{2}}}dxdy And I know the answer SHOULD be \pi Homework Equations Polar-->rectangular identities maybe? x--> rcos, y--> rsinθ, dxdy--> rdrdθ The Attempt at a Solution I tried using polar coordinates, but it...

∫∫cos(x^2 + y^2)dA, where R is the region that lies above the x-axis within the circle x^2 + y^2 = 9. Answer: .5pi*sin(9) My Work: ∫(0 ->pi) ∫(0 -> 9) cos(r^2) rdrdθ u = r^2 du = 2rdr dr = du/2r .5∫(0 ->pi) ∫(0 -> 9) cos(u) dudθ .5∫(0 ->pi) sin(u)(0 -> 9) dθ .5∫(0 ->pi)...

Good day, all: We recently hit double/triple integrals in my multivariable calculus course and I have found that my integration abilities are, well, *beyond* rusty ... and so the problem below, which is one of the very first on my current problem set, has me stumped. Homework Statement...