Double integral Definition and 558 Threads

Homework Statement Hi there, I've got this doubt about a double integral. I have this region: \displaystyle\int_{-1}^{2}\displaystyle\int_{-\sqrt[ ]{4-x^2}}^{1-x^2}f(x,y)dydx And the thing is, how this region would look like? Would it look like this?: The thing is that after the cut...

\int_{1}^{2}\int_{y}^{y^3}e^{\frac{x}{y}}dxdy \int_{1}^{2}ye^{y^2}-eydy taking u = y^2 \frac{1}{2}\int_{1}^{4}e^udu-\int_{1}^{2}eydy e^4/2-2e

http://img844.imageshack.us/img844/3293/17693169.jpg I follow it up to the third step, but how did they get the bottom denominators of -4? shouldn't it be -1?

Homework Statement Evaluate the double integral. [I don't know how to write it in latex, sorry, but its the double integral over the region 'd' of ye^x dA D is the triangular region with vertices (0,0), (2,4), and (6,0). Homework Equations The Attempt at a Solution So the...

Homework Statement Here are my problems.[PLAIN]http://img256.imageshack.us/img256/2254/whatua.jpg Homework Equations possible conversions to polar coordinates but I doubt that's needed. Fubini's theorem?The Attempt at a Solution So for the h(x,y) integral does not exist, I got this: i...

Homework Statement Okay here's the problem: Consider the region R interior to a circle(of r =2) and exterior to a circle(r=1). 1.Using cartesian coords and double integral, calc the area of annulus. 2. repeat calculation above but using double integral with polar coords The...

Homework Statement Homework Equations The Attempt at a Solution Please tell me if I am wrong. I suspect about the ranges. Are my range corrrect?

Homework Statement Use double integrals to find the volume of the region in the first octant (x, y, z all more than or equal to zero) bounded by the vertical plane 2x + y = 2 and the surface z = x2 Homework Equations The Attempt at a Solution I'm having major problems visualizing...

Homework Statement See Figure. Homework Equations N/A The Attempt at a Solution Simplifying the double integral, \int \int_{R} \sqrt{1 + 4x^2 + 4y^2} dA Am I suppose to put in the bounds for part a, as part of simplifying the integral? This brings me to part b along with...

Hi there! I am having a bit of a trouble when I try to work out a demonstration involving Dirac delta functions. I know, they are not real functions, and all that, but it only makes my life more difficult :) Lets begin by the beginning to see if anyone can help. The first equation I will...

just wondering if i can still do this, attempted the following: ʃʃ cos(x+y)dxdy with upper limits of pi/2 and lower limits of 0 for both integrals My answer came out as 0. Can anyone confirm this?

i am confused about the double integral ʃʃ cos(x+2y)dA, where R = [0,pi]x[0,pi/2] i realize for the integral that i must do u-substitution. when i do this, however, do i also have to change the boundary conditions as in a single integral? i got -8 without changing the boundary conditions...

If: x = f(t) (continuous and differentiable) y = g(t) (continuous) x is nondecreasing on [a, b] y is nonnegative on [a, b] Then when we trace the points (x,y) from t=a to t=b, we can calculate the area bounded above by the traced curve (below by y = 0, left by x = f(a), and right by x = f(b))...

Homework Statement With a > 0, b > 0, and D the area defined by D: \frac{x^2}{a^2} + \frac{y^2}{b^2} \leq 1 Change the integral expression below: \iint\limits_D (x^2+y^2) dx\,dy by using x = a r cos θ, y = b r sin θ. After that evaluate the integral. The Attempt at a Solution...

hey, i'm having some difficulties solving a problem. i want to know exactly how to go about solving it, since i am studying for a final exam. i know that i need to change the order of integration, but i'd also like to see how it's done correctly, since no official answers are provided... (my...

double integral to single by "magic" substitution Hi, I have a double (actually quadruple, but the other dimensions don't matter here) integral which looks like this: \iint_0^\infty \frac{d^2 k}{k^2} Now, someone here told me to replace that with \int_0^\infty \frac{1}{2} 2\pi...

Homework Statement Set up and evaluate a double integral to find the volume of the solid bounded by the graphs of the equation x2+y2+z2=r2 Homework Equations Not much equations, just setting the integral up, however I have no idea. The Attempt at a Solution I know...

Homework Statement I=\int^{a}_{0}dx\int^{\sqrt{a^{2}-x^{2}}_{0}(x-y)dy Homework Equations r^{2}=x^{2}+y^{2} The Attempt at a Solution Im thinking that the question is asking to integrate the first quarter of the circle of radius a between 0 and pi/2. In that case I've changed...

Homework Statement See attachment. Change the Cartesian integral into an equivalent polar integral, then evaluate the integral. I have no problems at all converting the actual function I am integrating or the integration itself, it is just the limits I cannot do. I've posted two...

Homework Statement Use polar coordinates to find the volume of the given solid: Under the cone z = Sqrt[x^2 + y^2] Above the disk x^2 + y^2 <= 4 2. The attempt at a solution I tried using formatting but I couldn't get it right so I'll explain...I changed variables by making the upper and...

Homework Statement So I have to use the type I type II region formula to find the volume under the equation (2x-y) and over the circular domain with center (0,0) and radius 2. Do I have to split this circle into semi-circles and treat it as 2 type I domains? I got the following limits for...

So I have to use the type I type II region formula to find the volume under the equation (2x-y) and over the circular domain with center (0,0) and radius 2. Do I have to split this circle into hemispheres and treat it as 2 type I domains? I got the following limits for the top half, but I get...

Homework Statement Eveluate by reversing order of integration \int^{2}_{0}\int^{1}_{y/2} ye^{x^3}dxdy Homework Equations The Attempt at a Solution this is what I got... \int^{1}_{0}\int^{2x}_{0} ye^{x^3}dydx I end up with... \int^{1}_{0} 2x^2e^{x^3}dx I don't know...

Homework Statement \int_{0}^{1} \int_{0}^{1} \sqrt{4x^2 + 4y^2 + 1} dx\,dy The Attempt at a Solution This integral is tough for me, I couldn't think of a way to evaluate it. Can you suggest me the first step to do this problem? Thanks!

Homework Statement Change the order of integration and evaluate the following double integral: I = {\int_0^{1} \left({\int\limits_{y}^{1} 30 y\sqrt{1+x^3} \mathrm{d}x }\right) {\mathrm{d}y} So thenn i did = 30 \int_0^{1} \sqrt{1+x^3} \left({\int_0^{x} y \mathrm{d}y}\right)...

Homework Statement http://img687.imageshack.us/img687/6092/dvojniinteg3.png Uploaded with ImageShack.us The Attempt at a Solution =k^2*X^2*a^3/6 Is this the correct solution?

Homework Statement trying to evaluate the double integral from 0 to infinity and 0 to infinity of [(x^2 + y^2)/1 + (x^2-y^2)^2]e^-2xy dxdy using the coordinate transformation u=x^2-y^2 and v=2xy Homework Equations The Attempt at a Solution so i calculated the jacobian...

Homework Statement Looks like I'm back with another question already :frown: I need to change the order of integration for this double integral and then evaluate it, but I get to a point where I'm not sure what to do. Homework Equations \int^3_{0} \int^9_{y} \sqrt{x}cos(x) dx dy The...

Homework Statement I'm trying to model the potential field in and around a symmetrically charged disc where the charge density drops exponentially from the center. Homework Equations This can be done by solving the double integral: \int ^{2 \pi} _{0} \int ^{\infty} _{0} \frac{r e^{-r/b}...

Find the volume under the cone z = sqrt ( x2+y2 ) and on the disk x2+y2 < 4. Use polar coordinates. Graphing x2+y2 < 4, I get a circle centered at 0,0 with radius of 2 So theta goes from 0 to 2pi Also, since x2+y2 < 4 This means that r^2 < 4 so -2 < r < 2...

Homework Statement Evaluate the following double integral: ∫ ∫ R sin (x/y) dA where R is the region bounded by the y axis, y=pi and x=y^2 Homework Equations as in problem statement The Attempt at a Solution Well I started this question by drawing the area to be evaluated...

Homework Statement \displaystyle\int\int\sqrt{4-x^2-y^2} dA R{(x,y)|x^2+y^2\leq4 .. 0\leq x} The Attempt at a Solution So far i have: \displaystyle\int^{\pi}_{0}\int^{r}_{0}\sqrt{4-r^2} rdrd\theta Solving i get...

Homework Statement integrate f(x,y) = sqrt(x^2+y^2) over triangle with vertices (0,0) (0,sqrt2) (sqrt 2, sqrt 2) Homework Equations x= rcosO, y = rsinO x^2+y^2=r^2 The Attempt at a Solution im supposed to use a double integral converted to polar coordinates, so i used...

Homework Statement Ok so I solved the problem, I think. I would just like to check my work. So the problem is: Use polar coordinates to find the volume of the given solid bounded by the paraboloids z = 3x^2 + 3y^2 and z = 4 - x^2 - y^2. Homework Equations r^2 = x^2 + y^2 x = r cos...

Homework Statement \displaystyle\int^1_0 \int^{e^x}_{1}dydx Homework Equations noneThe Attempt at a Solution the above integral i can do with no problem, but changing the order of integration give me a totally different answer and need to know if i am doing it correct...

Homework Statement Evaluate the integral. 1|0 s|0 ( t . sqrt ( t2 + s2 ) dt dsI hope the way I've written it makes some sort of sense. The Attempt at a Solution After getting my head around changing the order of integration I get hit with this question and for some reason am totally...

Homework Statement Evaluate the volume under z^2 = x^2 + y^2 and the disc x^2 + y^2 < 4. Just wondering what I should write to constitute a proper solution. Would this do?: V=(int)(int) z dA R is {x²+y² < 4} [context: R in other problems was the region over which integrals were performed]...

Homework Statement Evaluate \int\int x^{2}e^{x^{2}y} dx dy over the area bounded by y=x^{-1}, y=x^{-2}, x=ln 4 Homework Equations The Attempt at a Solution \int^{1}_{(ln 4)^{-2}}\int^{y^{-1}}_{y^{\frac{-1}{2}}}x^{2}e^{x^{2}y}dx dy I got this far before I realized that this wasn't a...

[sloved]reversing order of integration of double integral qns. Homework Statement pls refer to attached picture. Homework Equations The Attempt at a Solution intially upper and lower limits are , x^2 < y< x^3 and -1<x<1 sketched y=x^2 and y= x^3. => sqrt(y) =x and cube root...

Homework Statement Convert to polar coordinates to evaluate \int^{2}_{0}\int^{\sqrt(2x-x^2)}_{0}{\sqrt(x^2+y^2)}dydxThe Attempt at a Solution Really I'm just not sure how to convert the limits of integration. I know \sqrt(2x-x^2) is a half-circle with radius 1, but I'm not really sure where...

Homework Statement Use polar coordinates to find the volume of the solid enclosed by the hyperboloid -x^2-y^2+z^2=1 and the plane z=2. The Attempt at a Solution Solving for z of the equation of the hyperboloid I find z = Sqrt(1 + x^2 + y^2). Letting z = 2 to determine the curve of...

Homework Statement Let Ω ⊂ R^2 be the parallelogram with vertices at (1,0), (3,-1), (4,0) and (2,1). Evaluate ∫∫_Ω e^x dxdy. Hint: It may be helpful to transform the integral by a suitable (affine) linear change of variables. Homework Equations The Attempt at a Solution Ok...

Homework Statement Let Ω ⊂ R^2 be the parallelogram with vertices at (1,0), (3,-1), (4,0) and (2,1). Evaluate ∫∫_Ω e^x dxdy. Hint: It may be helpful to transform the integral by a suitable (affine) linear change of variables.Homework Equations The Attempt at a Solution Ok here is what I have...

Homework Statement Evaluate \int\intT (x^2+y^2) dA, where T is the triangle with the vertices (0,0)(1,0)(1,1) Homework Equations The Attempt at a Solution \int d\theta \int r^3 dr Thats how far I got, not really sure about boundries on r. First integrals boundrie should be 0 to pi/4. Is...

\int_{0}^{\infty}fdx\int_{\frac{x-tx}{t}}^{\infty}dy=\int_{0}^{\infty}dx\int_{\frac{x-tx}{t}}^{\infty}fdy f is a function of x and y can i move f like i showed? can i change the order of integration ?

Homework Statement By transforming to polar coordinates, evaluate the following: \int^{a}_{-a}\int^{\sqrt{}{{a^2}-{x^2}}}_{-\sqrt{{a^2}-{x^2}}}dydx Homework Equations The Attempt at a Solution I can get the right answer to this but only after guessing that the inner limits...

Homework Statement sketch the region of integration, and evaluate the integral by choosing the best order of integration \int^{8}_{0}\int^{2}_{x^{1/3}}\frac{dydx}{y^{4}+1} Homework Equations integration by parts The Attempt at a Solution after sketching the graph and changing the...

Hi, I am actually not really concerned about what the whole details are but more whether my approach is correct to show the following statement: Let f be continuous on a closed bounded region \Omega and let (x_0 ,y_0) be a point in the interior of \D_r. Let D_r be the closed disk with center...

1. Homework Statement [/b] Use the transformation that takes the unit square to a triangle to compute the integral \int\int_{B}2x+3y dA Where B is a triangular region with vertices (0,0), (5,2), and (3,4). The Attempt at a Solution What I did was I drew the region on an xy...

what'd I do wrong? I was told I didn't include the bound y<=x but that still hasn't helped me figure out where I miss stepped thanks -Ben