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.
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!!
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...
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
$$ m= \int_{E}...
Homework Statement
A metal wire is given a ceramic coating to protect it against heat. The machine that applies the coating
does not do so very uniformly.
The wire is in the shape of the curve
The density of the ceramic on the wire is
Use a line integral to calculate the mass of the...
Homework Statement
A sphere has a diameter of ##D = 2\rho = 4cm##. A cylindrical hole with a diameter of ##d = 2R = 2 cm## is bored through the center of the sphere. Calculate the volume of the remaining solid. (Spherical or cylindrical coordinates?)
hint: Place the shape into a convenient...
Homework Statement
Given
E = [(x,y,z) s. t. 0 \leq x \leq 2, 0 \leq y \leq \sqrt{2x - x^2}, 0 \leq z \leq 2]
Calculate
\int_E z^3\sqrt{x^2+y^2}dxdydz
Homework Equations
In cylindrical coordinates:
x=rcos(\theta)\\y=rsin(\theta)\\z=z\\dxdydz = \rho d\rho d\theta dz
The Attempt at a...
Homework Statement
Find the triple integral for the volume between a hemisphere centred at ##z=1## and cone with angle ##\alpha##.
The Attempt at a Solution
What I tried to do first was to get the radius of the hemisphere in terms of the angle ##\alpha##. In this case the radius is ##\tan...
Homework Statement
Calculate the volume integral of the function $$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...
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...
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...
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...
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...
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...
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...
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...