Triple integral Definition and 16 Discussions

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



{\displaystyle \mathbb {R} ^{2}}
(the real-number plane) are called double integrals, and integrals of a function of three variables over a region in



{\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. mncyapntsi

    Confused about polar integrals and setting up bounds

    So I want to subtract the two surfaces, right? I really don't know where to start... I am guessing this would be some sort of triple integral, however I am very confused with the bounds. Any help would be greatly appreciated! Thanks!!
  2. Draconifors

    Finding center of mass of solid

    Homework Statement A solid B occupies the region of space above ##z=0## and between the spheres ##x^2 + y^2 + z^2 = 16## and ##x^2+y^2+(z-1^2) = 1##. The density of B is equal to the distance from its base, which is ##z = 0##. The mass of the solid B is ##\frac{188\pi}{3}##. Find the...
  3. T

    Mass of Region Bounded by y=sin(x), z=1-y, z=0, and x=0

    Homework Statement On a sample midterm for my Calc 3 class the following question appears: Find the mass of (and sketch) the region E with density ##\rho = ky## bounded by the 'cylinder' ##y =\sin x## and the planes ##z=1-y, z=0, x=0## for ##0\le x\le\pi/2##. Homework Equations $$ m= \int_{E}...
  4. defaultusername

    Mass of Ceramic on a Wire

    Homework Statement A metal wire is given a ceramic coating to protect it against heat. The machine that applies the coating does not do so very uniformly. The wire is in the shape of the curve The density of the ceramic on the wire is Use a line integral to calculate the mass of the...
  5. TheSodesa

    A sphere with a hole through it (a triple integral).

    Homework Statement A sphere has a diameter of ##D = 2\rho = 4cm##. A cylindrical hole with a diameter of ##d = 2R = 2 cm## is bored through the center of the sphere. Calculate the volume of the remaining solid. (Spherical or cylindrical coordinates?) hint: Place the shape into a convenient...
  6. G

    Another triple integral problem

    Homework Statement Calculate \int_D \frac{dxdydz}{\sqrt{x^2+y^2+(z-A)^2}}, \: A>R on ## D = {(x,y,z)\: s.t. x^2+y^2+z^2 \leq R^2}##. Homework Equations In spherical coordinates: x=\rho cos\theta sin\phi\\y=\rho sin\theta sin\phi\\ z=cos\phi\\dxdydz=\rho^2sin\phi d\theta d\rho d\phi The...
  7. G

    Triple integral problem

    Homework Statement Given E = [(x,y,z) s. t. 0 \leq x \leq 2, 0 \leq y \leq \sqrt{2x - x^2}, 0 \leq z \leq 2] Calculate \int_E z^3\sqrt{x^2+y^2}dxdydz Homework Equations In cylindrical coordinates: x=rcos(\theta)\\y=rsin(\theta)\\z=z\\dxdydz = \rho d\rho d\theta dz The Attempt at a...
  8. P

    Volume of ice cream cone triple integral

    Homework Statement Find the triple integral for the volume between a hemisphere centred at ##z=1## and cone with angle ##\alpha##. The Attempt at a Solution What I tried to do first was to get the radius of the hemisphere in terms of the angle ##\alpha##. In this case the radius is ##\tan...
  9. physkim

    Volume integral of a function over tetrahedron

    Homework Statement Calculate the volume integral of the function $$f(x,y,z)=xyz^2$$ over the tetrahedron with corners at $$(0,0,1) (1,0,0) (0,1,0) (0,0,1)$$ Homework Equations I was able to solve it mathematically, but still can't figure out why the answer is so small. I only understand...
  10. P

    Finding centroid of a solid

    Homework Statement Use cylindrical coordinates to find the centroid of the solid. The solid that is bounded above by the sphere x2 + y2 + z2 = 2 and below by z = x2 + y2 Homework Equations x = rcos(theta) y= rsin(theta)[/B] The Attempt at a Solution I am having trouble trying to find...
  11. T

    Volume inside a sphere and cone

    Find the volume laying inside x^2 + y^2 + z^2 =2z and inside z^2 = x^2 + y^2. This is a problem my professor made, so I have no way of checking my answer. What I did first was completed the square for the sphere and got x^2 + y^2 + (z-1)^2 = 1, which is a sphere of radius one shifted above the...
  12. AdityaDev

    Acceleration due to gravity at the centre of a hemisphere

    Homework Statement find the acceleration due to gravity at the centre of a solid hemisphere. Homework Equations ##F=\frac{GMm}{r^2}## The Attempt at a Solution i decided to go for cylindrical coordinayes (which is way beyond my syllabus). I did some research though. let me take a point...
  13. V

    Divergence Theorem Question (Gauss' Law?)

    If F(x,y,z) is continuous and for all (x,y,z), show that R3 dot F dV = 0 I have been working on this problem all day, and I'm honestly not sure how to proceed. The hint given on this problem is, "Take Br to be a ball of radius r centered at the origin, apply divergence theorem, and let the...
  14. K

    Triple Integral of a cone bounded by a plane.

    Homework Statement find the volume using spherical coordinates of the region bounded above by z=9 and below by z=sqrt(x^2+y^2) in the first octant. Homework Equations The Attempt at a Solution I found this volume using cartesian and cylindrical coordinates, so I know the answer I am looking...
  15. K

    Triple Integral, Volume of an Egg

    Homework Statement I need to find the volume of an egg with a shape described by: z = 1/2(x2 + y2) and z = 6 - x2 - y2 I am also given that the egg is 6cm in length. Homework Equations I roughly graphed the two surfaces. The first being paraboloid that opens up from the origin, and the...
  16. N

    Triple Integral Meaning

    Okay so I just have a question on triple integrals. I understand how to use triple integrals to find volumes, but what I really don't understand is what I am really getting when I take the triple integral OF a function. I understand physical examples like taking the triple integral of a...