Double integral Definition and 558 Threads

Homework Statement A cylindrical drill with radius r1 is used to bore a hole throught the center of a sphere of radius r2. Find the volume of the ring shaped solid that remains. Homework Equations x=r*cos(theta) y=r*sin(theta) The Attempt at a Solution i know that the boundaries...

Homework Statement A region R is given. (ill just tell you that it is a triangle, given by lines x = -2, y = 2, and y = x). Decide whether to use polar coordinates or rectangular coordinates and write \int\int f(x,y)dA as an iterated intergal, where f is an arbitrary continuous function...

The double integral xcosy is bounded by y=0, y=x^2, and x=1. I was able to integrate almost wholly through; however, toward the end I was unsure what to do when i was asked to plug in x^2 into x^2. What do I do?! Here is an image of my work on the white board. Please, if my hand writing is...

Homework Statement I wish to find the following integral over the rectangle [-a,a] in u and [-b,b] in v using Mathematica. The constants a and b are positive (and non-zero). The variables x and y are in Reals. A(x,y)=\int_{-a}^a{\int_{-b}^b{\log{\left[(u-x)^2+(v-y)^2\right]}{\rm d}v}{\rm...

Homework Statement I have the bounds, 0≤y_{1}≤2, 0≤y_{2}≤1, and 2y_{2}≤y_{1}. I now have a line u=y_{1}-y_{2} and I'm trying to find the area such that y_{2}≥y_{1}-u. The integral comes down to two parts, the first of which I'm stuck on (when 0≤y1≤1). I'm pretty sure I have one way setup...

My idea was that the limits are and that the anti-derivative of dy was xlog(1+y^2) but that seems wrong... maybe use these limits instead and start with dx? gives us then we take dy guess, i figured it out eventually with the help of wolfram with the last integration

find the area of the double integral ∫∫x + y (is the triangular vertices (0.0) , (2,2) and (4,0)) how to find the values of x and y.

∫u=3 and l=0 u= x and l= 0∫ (x2 + y2 )dydx solve by reversing the order of integration. u and l means upper and lower limit. this is a double integral by the way. i don't understand how the limits are found when reversing the order and the idea of diagrams. please help me

Homework Statement Evaluate f(x,y)=y2\sqrt{1-x2} over the region x2+y2< 1 Homework Equations The Attempt at a Solution using x limits between -1 & 1 followed by the y limits of 0 & \sqrt{1-x2} \int\inty2\sqrt{1-x2}.dy.dx Evaluating this and multiplying be 2 to get the...

Area in cardioid and outside circle -- Using Double Integral Homework Statement Find the area inside of the cardioid given by r = 1 + cos\theta and outside of the circle given by r = 3cos\theta. Homework Equations \int\intf(x,y)dA = \int\intf(r,\theta)rdrd\theta not really relevant...

Homework Statement Evaluate \int\intD(x+2y)dA, where D is the region bounded by the parabolas y=2x2 and y=1+x2Homework Equations dA = r*drd\vartheta r2=x2+y2 The Attempt at a Solution Well, I know I need to put D into polar coordinates, but I'm lost on this...

Homework Statement Find the volume under the sphere x^2+y^2+z^2=r^2 and above the plane z=a, where 0<a<r Homework Equations x^2+y^2+z^2=r^2 is the equation of a sphere with radius r centered at the origin z=a is the equation of a plane with height a parallel to the xy plane V = ∫∫z...

Link: http://imageshack.us/photo/my-images/39/18463212.jpg/ This is a very long problem so I drew it to make things simpler. Part a) tells me to set up a double integral in polar coordinates giving the total population of the city. I have the following: 2π...4 ∫...∫ δ(r, θ) r dr...

[b]1. Find the area of the ellipse (2x + 5y − 3)^2 + (3x − 7y + 8)^2 < 1 I have no idea what this looks like, and hence I can't figure out the limits. Maybe I could transform it into a more familiar form using a translation and rotation? Please help.

Homework Statement Use stokes theorem to find double integral curlF.dS where S is the part of the sphere x2+y2+z2=5 that lies above plane z=1. F(x,y,z)=x2yzi+yz2j+z3exyk Homework Equations stokes theorem says double integral of curlF.dS = \intC F.dr The Attempt at a Solution...

Homework Statement Invert the limits of integration of the following integrals: 1 ) \int_{0}^{4} dx \int_{0}^{x} f(x,y)dy \int_{0}^{2} dx \int_{0}^{\surd (4 - x^2)} f(x,y)dy \int_{0}^{1} dy \int_{y}^{2-y} f(x,y)dx These are 3 different integrals in 3 separate exercises, they're...

Homework Statement Solve the following Double Intergral, and show the answer is the same, regardless of which order you integrate. The integral is between the boundaries y=x and y=x^2 Homework Equations \int\int_R (x^2 + 2y)dxdy The Attempt at a Solution So first of all i integrated with...

Homework Statement integrate 1/((1+x^2+y^2)^2) dx dy Both x and y going from 0 to infinity Homework Equations x^2+y^2 =r The Attempt at a Solution After that I get 1/(1+r^2) ^2 Cannot visualize the function, do not know what the limits are. If I could have any help it...

Homework Statement Calculate the double integral over D \int\int x*ln(2x + y)/y^3 dx dy D is the finite area in the xy-plane within the straight lines 2x + y = 1 2x + y = 3 x = y x = 2y Homework Equations - The Attempt at a Solution I thought it was obvious to make the variable substitution u...

Homework Statement Use polar coordinates to change the following double integral to a single integral involving only the variable r. Double-Integral( \sqrt{1+(x^{2}+y^{2})^{2} ) The x-y region is x^2 + y^2 = 4 in the first quadrant. 2. The attempt at a solution I got upto this...

i have no idea how to use the functions on here to ill try my best. \int(upper bound a lower bound 0)\int(upper bound 0 lower bound -sqrt(a2-y2) of the function x2y.dxdy firstly trying to map it out... i think its the quarter circle in the top left quadrant with boundaries 0 to a along...

Homework Statement Integrate Double Integral of e^(x^4), First Bound- a = 3*(sqroot(y)) and b= 2 and 2nd Bound - C = 0 and D = 8? Homework Equations The Attempt at a Solution We have been working on this all day and we have tried changing bounds but we cannot find an solution that...

Please ignore this thread (problems with Latex and the question looks like a jumbled mess!)

Homework Statement If you know the area of a region with constant density, and you know xbar and ybar, then its possible to compute \int\int ax+by dA for any constant a and b. [Hint: write down the formulas for the center of mass of a region. If A=5 and (xbar,ybar)=(2,3), Compute\int\int...

Hello. Can anyone help me, please? R = { (x,y) \in R² | 0 \leq x \leq 1, 0 \leq y\leq 1-x} f is continuous at [0,1] Show that \iint_R f(x+y) dxdy = \int_{[0,1]} u f(u) du

Homework Statement Use the transformation x= \sqrt{v- u}, y = u + v to evaluate the double integral of f(x, y) = \frac{x}{(x^2 + y)} over the smaller region bounded by y = x^2, y = 4 − x^2, x = 1. Homework Equations The Attempt at a Solution d:={ (x,y)| -\sqrt{2}<x<1 , x^2<y<...

Double Integral bounded by Circle? Double integral of (2x-y)dA bounded by circle of radius 2, centered at origin I just need to figure out the limits for my integrals... I am basically lost, can someone show me how to break this up. I tried doing what I did with the previous triangle bound...

Homework Statement Using polar coordinates, evaluate the integral which gives the area which lies in the first quadrant between the circles x2 + y2 = 256 and x2 - 16x + y2 = 0. Homework Equations The Attempt at a Solution Finding the intervals of integration for the polar coordinates. From the...

Apologies for not being proficient enough in the use of Latex to write this problem properly I hope it will suffice if I simply describe it: It is the integration of f(x,y)=(sin(y))/(x+y) with respect to x between limits 0 and y which I've found to give ( sin(y) ) ( ln(2y) ) This...

I'm reading through a proof (the full theorem statement is at the bottom of the post) in a book on probability and I'm having trouble following a line in the proof. The line reads as follows: \int_{0}^{\infty} \int_{x:g(x)>y} f(x) dx dy = \int_{x:g(x)>0} \int_{0}^{g(x)} dy f(x) dx Where...

hi, i've never posted on here before but would appreciate any help given for this question. The scattering term (4th term) in the Boltzmann transport equation contains a double integral. What are you integrating over in each case and why is this? this may be a simple question but...

Homework Statement Outer Integral: From zero to one dy Inner Integral: from zero to y^2 dx Function is: 3y^3 * e^(xy) Homework Equations None The Attempt at a Solution Have tried numerous u substitutions on e^(xy), but taking me nowhere. I am clearly doing something wrong...

Hello everybody, I am having trouble doing this polar double integral. The problem says.. Find the area of the region.. \frac{1}{2}y^2 \leq x \leq 2y 0 \leq y \leq 8 It is hard for me to come up with the limits of integration. Checking the answer would be easy because I can...

Homework Statement int (1/(4-r^2)^0.5) dr dx, r=0 to 2sinx, x=0 to 2pi Homework Equations How to continue the integral of x The Attempt at a Solution I'm stuck at int(arcsin(sinx)) dx, x=0 to 2pi

I've been working on a problem involving a large meteoroid passing over the Earth and what its gravitational effects would be on the Earth's mantle. I developed an equation for this, and I've worked it down to a certain point, but unfortunately, I'm not sure how to finally solve it. By the way...

Evaluate \int\int xy dxdy where D is the triangular region {(x,y) element of R2| x+y <= 3, x >=0, y >= 0} ( You have to work out the limits of the integrals from the region D)the bit i get confused about in these questions are the limits of the integrals So i just want to check my answer. I...

Homework Statement ∫4to5 ∫1to2 (1x + y)−2 dy dx The Attempt at a Solution I am confused about what to do with this negative 2. Any ideas?

Recently, I've been working on a program to simulate diffuse light, and I've hit a snag. I need to solve (at least so that a computer can compute L(x) quickly) something of the form: L(x)=T(x)+c\int_0^{l_2}\int_0^{l_1} W(x,u_1,u_2) L(u_1) du_1 du_2 W and T are pretty well behaved, and...

Homework Statement Let D be the triangular domain given by 0\leq y \leq3, (y/3)-1 \leq 1-(y/3). Then \int\int (e-x^{5}e^(sqrt(1+y^2)) Homework Equations The Attempt at a Solution There is a quick way to solve it by breaking apart the double integral and then, apparently the x^5...

Homework Statement I want to use polar coordinates to integrate 1/sqrt(x^2+y^2) dydx with limits of integration 0 < y < x and 0 < x < 3Homework Equations x=rcosO y=rsinOThe Attempt at a Solution I know that the area being integrated over is the triangle enclosed by y=x and x=3. I have my...

Homework Statement \int_{y=-infinity}^{infinity} \int_{x=-infinity}^{infinity} (x^4+y^4)/(1+x^2+y^2)^4 dx dy Homework Equations i'm not sure what the new limits are after the transformation to polar coordinates and how to solve the integral. The Attempt at a Solution i have my...

Homework Statement \int\intmin(x,y)*dx*dy both limits are from 0 to 1 what is the general solution of this question ? Homework Equations The Attempt at a Solution I attempted to solve by saying that x<y, finding the integral and then, y<x and finding its integral. meaning...

Homework Statement I was trying to find the volume of the intersection between 3 cylinders x^2 + y^2 = 1, y^2 + z^2 =1, and z^2 + x^2 =1. I set up the double integral in two different ways: 8\int_{\theta = \frac{-\pi}{4}}^{\frac{\pi}{4}}\int_{r = 0}^{1} \sqrt{1 - r^{2}\: cos^{2}\, \theta}\...

Homework Statement My homework problem is the double integral of y/1+xy dxdy. It is a definite double integral and both integrands have the values of a = 0 and b = 1. Homework Equations Integration by parts: uv - int(vdu) The Attempt at a Solution My first step of the double integral is I...

Homework Statement Compute the double integral where the region is a triangle with vertices (1,1),(2,4),(5,2).(please see the attachment) Homework Equations double integral((16-2x-3y)1/11)da The Attempt at a Solution first i found the equation of the three lines ...

Homework Statement Calculate the following: integrate [ (e^(x/y)) dy y=x^(1/3)..x ] dx x=1..8Homework Equations The Attempt at a Solution In this current state the dy part is not integrable, so there must be some trick with changing the variables or something else. Couldn't find a proper...

Evaluate the integral by reversing the order of integration. I'm not exactly sure how to approach this problem. Any help would be appreciated!

Homework Statement Can anybody prove the following double integral identity? How?: \int_{0}^{1} s(1-s) f(sx) ds = \int_{0}^{1} s^2 \int_{0}^{1} t f(tsx) dt ds Here f(x) is an arbitrary Riemann-integrable function. Thanks in advance. Homework Equations I've found the following but it...

Homework Statement question 2: http://www.math.ubc.ca/~haber/courses/math253/Welcome_files/asgn5.pdf" The Attempt at a Solution So for part a) I tried to plot my domain of integration and ended up concluding it was an area bounded by y=0, y=4-x^2, and x=1. Is this okay? In not too...

Homework Statement Hi there. I have this problem with double integral, which says: calculate using a double integral the volume limited by the given surfaces: x^2+y^2=4, z=4-y z=0 Its a cylinder cut by a plane. At first I've did this integral...