Double integral Definition and 573 Threads

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.

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  1. W

    Double Integral evaluation help

    Homework Statement Evaluate the double integral \int \int_{R} ln(xy) dA where R is the rectangle bounded by x=e, x=e^2,y=1,y=e. Homework Equations ln (xy) = ln x + ln y The Attempt at a Solution I was just wondering, do I need to do anything other than take the integral with respect to x...
  2. B

    Double Integral of (x+y) over Region Bounded by y=x^2 and y=x^(1/2)

    Homework Statement \int \int_R (x+y) dA R is the region y = x^2 and y = x^(1/2) Homework Equations I've sketched the graph and the functions are equal at (0,0) and (1,1) The Attempt at a Solution Would the limits of the integral be: \int_{0}^{1) \int_{0}^{1} (x+y) dydx...
  3. A

    Using the double integral to find the volume bounded betwee two solids

    Homework Statement Find the volume of the solid T enclosed above by the sphere x^2+ y^2 + z^2 = 2 and below by the parabloid x^2 + y^2 = z Homework Equations The double integral. Possiblly polar coordinates (x = r*cos(theta) y = r*sin(theta)). z = f(x,y) The Attempt at a Solution...
  4. J

    Cartersian double integral converted to polar and then evaluated #2

    Homework Statement convert line one to polar integral and then evaluate see problems 16 attachement Homework Equations r^2=y^2+x^2 The Attempt at a Solution I changed to polar and evaluated the double integral but I come up with an answer of negative pi which seems odd since it...
  5. C

    Integration by Parts of a Double Integral

    Homework Statement ∫∫xy(x^2+y^2)^(1/2)dydx over the range 0 to 1 for both x and y. Homework Equations I believe that it requires integration by parts. Any help would be greatly appreciated.
  6. D

    Double Integral - Polar Coordinates

    Homework Statement Evaluate by changing to polar coordinates Homework Equations Can't figure out how to make the integral stop after the sqrt(9-x^2) \int_0^\frac{3}{\sqrt(2)} \int_x^{\sqrt(9-x^2)} e^-(x^2+y^2) dy dx The Attempt at a Solution I'm not sure where to really start on this one...
  7. K

    How do I evaluate a double integral over a specific region?

    Homework Statement Sketch the region of integration and then evaluate the double integral: Homework Equations \int\intx2exydA over the region R= {(x,y), y<=x<=1, 0<=y<=1} The Attempt at a Solution I have managed to do half of the problem and integrate it respect to x but then...
  8. C

    Double integral showing a range of values?

    Hi all, I am faced with this question. I am asked to show that 2(\sqrt{5}-2)\pi\leq\iint_{R}\frac{1}{\sqrt{4+sin^2x+sin^2y}}dA\leq\frac{\pi}{2} Noting that the double integral is to be performed on region R which is bounded by the circle x^2+y^2=1 From what I know, the double...
  9. C

    How to Calculate the CDF of e^-y Using Double Integrals?

    Homework Statement \int^x_0\int^y_x e^{-v} dv du where u and v are just "dummy variables" Homework Equations The Attempt at a Solution \int^x_0\int^y_x e^{-v} dv du = \int^x_0 -e^{-y} + e^{-x} du = (-e^{-y} + e^{-x})x have I made a mistake somewhere?
  10. W

    Double integral in polar form: how do you find the boundaries?

    Hi I need to use a double integral to find the area of the region bounded by: r = 3 + 3sinQ where Q = theta. I know the bounds of the inner integral are from 0 to 3 + 3sinQ. However, I do not know how to determine the bounds of the outer integral. Any help would be greatly appreciated.
  11. D

    Calculating the Area of a Tetrahedron with Double Integral Calculus

    Consider the tetrahedron which is bounded on three sides by the coordinate planes and on the fourth by the plane x+(y/2)+(z/3)=1 Now the question asks to find the area of the tetrahedron which is neither vertical nor horizontal using integral calculus (a double integral)? I think they mean...
  12. V

    Double integral transformation

    Homework Statement evaluate the integral \int\int(x^4-y^4)e^{xy}dA where R is the region bounded by xy=1, xy=2, x2-y2=1, and x2-y2=4 Homework Equations The Attempt at a Solution This is my first time on the forum, so forgive me if there are mistakes in this post. I am...
  13. S

    Double integral by changing order and limits

    Homework Statement Evaluate the following double integral by changing the order of integration: ∫(lower 0 and upper 1)∫ (lower √x and upper 1) sin(((y^3)+1)/2) dydx [b]2. Homework Equations In case it's not clear from above! y is between √x and 1, and x is between 0 and 1. The...
  14. D

    Double integral of piecewise function

    Homework Statement Let f(x,y)= 1 if x is rational, 2*y if x is irrational Compute both double integrals of f(x,y) over [0,1]x[0,1] Homework Equations The Attempt at a Solution I'm tempted to say that we can do the dydx integral since when x is rational, integrating y gives...
  15. A

    Some questions about double integral and vector calculus

    I've written all the questions in the PDF file ... Please help me !
  16. C

    Double Integral with Polar Coordinates

    Homework Statement \int^{0}_{-3}\int^{\sqrt{9 - x^2}}_{- \sqrt{9 - x^2}} \sqrt{1 + x^2 + y^2} dy dx Homework Equations x = rcos(theta) y = rsin(theta) The Attempt at a Solution By making \sqrt{9 - x^2} = y then changing it to polar coordinates, I got r to be +/-3 but I'm...
  17. C

    Determining Volume using double integral

    Homework Statement Find the volume of the solid in the first octant by the planes z = x + y + 1 and z = 5 - x - y Homework Equations The Attempt at a Solution How would I set this up?
  18. C

    Find Center of Mass for Double Integral in Region R

    Homework Statement A lamina occupies the region inside the circle x^2 + y^2 = 2y but outside the circle x^2 + y^2 = 1. Find the center of mass if the density at any point is inversely proportional to its distance from the origin. Homework Equations Xcm = double integral of y*f(x,y)...
  19. A

    Double integral polar coordinates trouble

    Homework Statement Consider the volume of a solid bounded by the cone: z = sqrt(x^2 + y^2) and the top half of the sphere x^2 + y^2 + z^2 = 18 that is for z >= 0 Using cylindrical coordinates, express the volume as a double integral. Homework Equations easy to sketch.. we can...
  20. A

    Xy coordinates to polar coordinates for double integral. hepl please

    Homework Statement ok change the region R = { (x,y) | 1 <= X^2 + y^2 <= 4 , 0 <= y <= x } to polar region and perform the double integral over region R of z=arctan(y/x)dA Homework Equations r^2 = x^2 + y^2, x = r*sin(@), y = r * cos (@) The Attempt at a Solution i got R = {...
  21. C

    What is the Probability of Double Bulb Failure within a Given Time Period?

    Homework Statement a) A lamp has two bulbs of a type with an average lifetime of 1000 hours. Assuming that we can modelthe probability of failure of these bulbs by an exponential density function \mu = 1000, find the probability that both of the lamps bulbs will fail within 100 hours. b)...
  22. S

    Integration over a Region in the First Quadrant without a Prefix.

    Hi, I was just wondering if the set up for this problem; integrate f(s,t)=e^slnt over the region in the first quadrant of the st-plane that lies above the curve s=lnt from t=1 to t=2 is: integral(t=1 to t=2)integral(s=ln1 to s=ln2) of e^slnt If that's not the right set up what am I...
  23. F

    Is This the Correct Setup for a Double Integral to Find Area Between Curves?

    simple question, for "find the area of the region between the xy-plane between the curves y=x3 and x=y2 where 0<x<1 , 0<y<1 is this the double integral for x3*y2 dy dx or for the double ingeral between x3+y2 dy dx?? i assume the first one? clarification needed please, can do the rest of the...
  24. C

    Double integral bounds after polar transformation

    Homework Statement Basically I'm just trying to convert a double integral into polar coordinates, but when I do it I get confused with my bounds. Homework Equations The Attempt at a Solution 4\int_0^{\infty}\int_0^{\infty}e^{-(u^2+v^2)}u^{2x-1}v^{2y-1}dudv (x and y are just numbers, not...
  25. C

    Simplifying a Double Integral: No Known Anti-Derivative

    Homework Statement The equation is \int^{t}_{0}e^{-t'\gamma}\int^{t}_{0}e^{t''\gamma}f(t'')dt''dt'. f(t) is a random function with no known anti-derivative. I need to simplify this into a single integral of one variable. Homework Equations above. The Attempt at a Solution I moved the...
  26. M

    Double Integral Limits for a Sideways Rectangle

    Homework Statement I have to solve the integral of (ytan^-1(x) - 3) inside the area of the rectangle with vertices (1,0), (0,1), (2,3), (3,2). How do I set up these limits? Homework Equations This is a tilted rectangle so I can't use just values for the limits? The Attempt at a...
  27. O

    How Does Using Dirac in a Double Integral Simplify the Integration Process?

    Hi, I have a question about using the dirac function in a double integral. Lets say you have the double integral over the two values x1 and x2: int( int( sin(x1) * dirac(x1-x2) * sin(x2) )) Does this just simplify to a single integral: int( (sin(x1))^2 ) thanks!
  28. S

    How do I reverse the order of integration for this double integral?

    Homework Statement Evaluate an iterated integral by reversing the order of integration \int^1_0\int^1_{y^2} ysin(x^2)dxdy Homework Equations The Attempt at a Solution I've got that the limits for x is between y^2 and 1, while the limits for y is between 0 and 1. Then I graphed...
  29. P

    Double Integral Volume Problem

    Homework Statement Find the volume of the solid enclosed by the cylinders z=x^2, y=x^2, and the planes z=0 and y=4. Homework Equations The Attempt at a Solution ∫∫ x^2 dA For the limits of integration, I obtained y=x^2 and y=4, x=0 and x=2 I changed the order of...
  30. D

    Polar Double Integral: Converting and Evaluating

    Homework Statement Given the integral: \int_1^2\int_{\frac{3}{\sqrt{x}}}^{\sqrt{3}x}{{(x^2+y^2)}^{\frac{3}{2}}}dy \; dx. Convert to polar and evaluate. Homework Equations r=\sqrt{(x^2+y^2)} The Attempt at a Solution Ok, I've gotten bounds on \theta, \frac{\pi}{6} \le \theta \le...
  31. C

    Double Integral in an Enclosed Region

    Homework Statement \int\int_B 1 dx dy where B is the region enclosed by x^2 + y^2 = 9? What if B is the region eclosed by y = x +3, y = 5 - x, and y=8 Homework Equations The Attempt at a Solution So for x^2 + y^2 = 9, does it mean that both x and y go from -3 to 3? bot...
  32. C

    How Do You Evaluate a Double Integral with Mixed Powers of x and y?

    Homework Statement Evaluate the following integral: \int \int_R x^3 y^4 dx dy Homework Equations The Attempt at a Solution I don't even know where to start. My professor just introduced us to double integrals, now there's a strike going on at my school, so classes are...
  33. S

    How do you solve the double integral of xe^(xy) bounded by x=0, y=1, x^2-y=0?

    How do you solve the double integral of xe^(xy)? bounded by x=0, y=1, x^2-y=0. I used both x-simple and y-simple methods but neither worked..I don't know what the limits are...thanks!
  34. M

    Double Integral Help - Solve Analytically

    can someone please give me some help on this integral. it should be solvable analytically. \int^{1}_{-1}\int^{2\pi}_{0}\frac{1}{2+x+\sqrt{1-x^2}cos(y)+\sqrt{1-x^2}sin(y)}dxdy
  35. F

    How do you solve the double integral

    Homework Statement How do you solve the double integral of cos(u/v) dudv, if the limits of u are v and -v, and the limits of v are 1 and 2? I tried doing it by parts but I didn't get it...
  36. CalleighMay

    Use a double integral to find volume of solids ?

    Hey guys! I have been on the forum for about a week or so and have compiled a lot of information and techniques to help me understand calculus, so i really appreciate everyone's help! I am a soon-to-be freshman in college and am taking a summer class, calculus II (took calc I in HS). This is...
  37. G

    Double integral with transformation

    Homework Statement \int\int x^2 dA where the area D is boundd by the ellipse 5x^2 + 4xy + y^2 = 1. Homework Equations The Attempt at a Solution I'm not sure where to start this question. A few ideas I'm exploring are: 1. rewrite in polar co-ordinates (not sure how to write an...
  38. G

    Double Integral: trouble manipulating algebra

    Homework Statement \int\int e^(^x^2^+^y^2^) dA where D is the region bounded by y = sqrt(1-x^2) and y = |x|. Homework Equations The Attempt at a Solution Obviously I can draw this region out and see what it looks like, and I will have to split the integral into two for negative...
  39. D

    Need help figuring out limits of double integral

    Homework Statement Apply Green's Theorem to evaluate this integral: Integral of: (6y + x) dx + (y + 2x) dy over the curve C, where C is: The circle (x - 2)^2 + (y - 3)^2 = 4 Homework Equations The Attempt at a Solution To use Green's Theorem for this I would need to figure...
  40. H

    Drawing the Domain for Double Integral of Function of Two Variable

    This is not for a specific problem, just in general: If we want to find the double integral of a function of two varaible, when we got other 1 varaible function to define the domain. What are the basics of drawing the domain in 2d (x, y only)?
  41. M

    How do I evaluate a double integral with a trigonometric function?

    Homework Statement Given the integral shown (in attachment), make a sketch of the region of integration, express the integral with the order of integration reversed and evaluate the integral after reversing the order of integration Homework Equations The Attempt at a Solution So...
  42. M

    Another double integral question

    Homework Statement Given the integral (shown in attachment) make a clear sketch of the region of integration and express the integral with the order of integration reversed. Evaluate the integral you found in (ii). Homework Equations The Attempt at a Solution Everything is in...
  43. D

    Substitution in double integral

    Homework Statement Use the transformation u = 3x + 2y and v = x + 4y to evaluate: The double integral of (3x^2 + 14xy + 8y^2) dx dy for the region R in the first quadrant bounded by the lines y = -(3/2)x + 1, y = -(3/2)x + 3, y = -(1/4)x, and y = (-1/4)x + 1. Homework Equations...
  44. A

    What is the result of this double integral?

    Consider the double integral \int_{-\infty}^{\infty}dx f(x) \, \int_{-\infty}^{\infty}dy g(y) The first one gives 0 the second one gives infinity (diverges). Then how to express the result of the integral? Is it 0 or infinity or neither (indeterminate)? Any other comments about the integration?
  45. M

    How do I choose the area of integration?

    Homework Statement Consider the integral shown in the sketch. Sketch the region of integration and express the integral with the reverse order of integration and evaluate it leaving your answer in surd form Homework Equations The Attempt at a Solution I shaded the area of integration but I...
  46. I

    Double Integral: Finding Triangle Area

    Homework Statement \int_a^b\,dx\int_a^x\,dy should give area of a triangle, I can’t see how. The Attempt at a Solution \int_a^b\,dx\,{(x-a)} but then I won’t get 1/2 (ab)… (don't get the first brackets in latex between the two x)
  47. K

    Ampere's law - single or double integral?

    Flux = Int B dA Why isn't this written as a double integral when we antidifferentiate over an areal?
  48. G

    Solving a Double Integral over a Rectangle with Given Vertices

    Evaluate \int\ \int_R\ x^2e^ydA Over the rectangle R with vertices (0,0), (1,0), (1,3) and (0,3). My answer: \int\ \int_R\ x^2e^ydA = \int_0^3\ \int_0^1\ x^2e^ydA = \int_0^3\ [x^3/3]_0^1 e^y dy = 1/3 \int_0^3\ e^ydy = 1/3 (e^3-1) Double integrals are new to me, so if...
  49. M

    Desperate college student needs help in double integral

    here is the problem I couldn't solve, anyone got any idea please help me. thank you very much in advance 1) use double integrals to derive the given formula for the volume of a right circular cone of radius R and height H. the volume of a cone is given by the formula (pi*R^2*H)/3 I...
  50. M

    Desperate college student needs help in double integral

    desperate college student needs help in double integral! here is the problem I couldn't solve, anyone got any idea please help me. thank you very much in advance 1) use double integrals to derive the given formula for the volume of a right circular cone of radius R and height H. the volume of...
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