Triple integral Definition and 316 Threads

Drawing : The drawing of the problem is shown alongside. This is also known as the problem of the "spherical cap". I have been able to do it by the normal way. It involves taking a thin slice of the liquid of thickness ##dz## at a height (or depth) ##z## from O and calculating its area ##A =...

In my textbook, a hint is the formula above, which can be used when we have a homogenous body. ##M## is the body's mass, but what does the remnant mean?

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!!

Greetings here is my integral Compute the volume of the solid and here is the solution (that I don't agree with) So as you can see they started integrating sinx from 0 to pi and then multiplied everything by two! for me sin(x) is an odd function and it's integral should be 0 over symmetric...

Good day here is the solution J just don't understand why the solution r=√2 has been omitted?? many thanks in advance best regards!

I am trying to solve it using cylindrical coordinates, but I am not sure whether the my description of region E is correct, whether is the value of r is 2 to 4, or have to evaluate the volume 2 times ( r from 0 to 4 minus r from 0 to 2), and whether is okay to take z from r^2/2 to 8

Let: \begin{align} r&=\sqrt{a^2 + p^2 - 2ap \cos \theta}\\ s&=a\\ t&=p\\ f(r) &= \text{continuous function of } r\\ g(s) &= \text{continuous function of } s\\ \end{align} Consider the expression: \begin{align} \int_{q'}^q \int_{b'}^b g(s)\ \int_{s-t}^{s+t} f(r)\ dr\ ds\ dt\ \end{align} We...

Let ## E=\left\{ (x,y,z) \in R^3 : 1 \leq x^2+y^2+z^2 \leq 4, 3x^2+3y^2-z^2\leq 0, z\geq0 \right\} ## - Represent the region E in 3-dimensions -represent the section of e in (x,z) plane -compute ## \int \frac {y^2} {x^2+y^2} \,dx \,dy \,dz## the domain is a sphere of radius 2 with an inner...

15.1.34 Evaluate $\displaystyle I=\int_{0}^{3\pi/2}\int_{0}^{\pi}\int_{0}^{\sin{x}} \sin{y} \, dz \, dx \, dy$ integrat dz $\displaystyle I=\int _0^{3\pi/2}\int _0^{\pi }\sin(y)\sin (x)\, dxdy $ integrat dx $\displaystyle I=\int _0^{3\pi/2}\sin \left(y\right)\cdot \,2dy$...

I would like to compute the triple integral of a function of three variables [FONT=Times New Roman]$f(x,y,z)$in R. I am using the package Cubature, Base, SimplicialCubature and the function adaptIntegrate(), Integrate and adaptIntegrateSimplex(). The integrand is equal to 1 only in certain...

Hello everybody. If anyone could help me solve the calculus problem posted below, I would be greatful. Task: Evaluate the moment of inertia with respect to Oz axis of the homogeneous solid A Bounded by area - A: (x^2+y^2+z^2)^2<=zSo far I was able to expand A: [...] so that I receive...

The integral is$$\int_0^4dz\iint xyz~dxdy$$Constricted to the quarter circular disk ##x^2+y^2=4## in the first quadrant. First I switched to polar coordinates and integrated the double integral by first writing it as:$$\int_0^4z~dz \int_0^\frac{\pi}2\int_0^2...

Homework Statement $$\int_{-23/4}^4\int_0^{4-y}\int_0^{\sqrt{4y+23}} f(x,y,z) dxdzdy$$ Change the order of the integral to $$\iiint f(x,y,z) \, \mathrm{dydzdx}$$What I have done It is just about: From ##x=0## to ##x=\sqrt{4y+23}## From ##z=0## to ##z=4-y## From ##y=\frac{x^2-23}{4}## to...

While deriving the volume of sphere formula, I noticed that almost everyone substitute the limits 0 to 360 for the angle (theta) i.e the angle between the positive x-axis and the projection of the radius on the xy plane.Why not 0to 360 for the angle fi (angle between the positive z axis and...

Hello all, I need to evaluate the following 3-dimensional integral in closed-form (if possible) \int_{y_1=0}^{\infty}\int_{y_2=0}^{\infty}\int_{x_2=0}^{zy_2}\exp\left(-\min(x_2,\,y_1(z-\frac{x_2}{y_2}))\right)e^{-(K-1)x_2}e^{-y_1}e^{-y_2}\,dx_2dy_2dy_1 where ##z## is real positive number, and...

Homework Statement Find the centre of mass of a uniform hemispherical shell of inner radius a and outer radius b. Homework Equations ##r_{CoM} = \sum \frac{m\vec{r}}{m}## The Attempt at a Solution Using ##x(r,\theta,\phi)## for coordinates...

Homework Statement Use cylindrical coordinates to evaluate triple integral E (sqrt(x^2+y^2)dv where E is the solid that lies within the cylinder x^2+y^2 = 9, above the plane z=0, and below the plane z=5-y Homework EquationsThe Attempt at a Solution So i just need to know how to get the bounds...

$\textsf{Find the volume of the given solid region bounded by the cone}$ $$\displaystyle z=\sqrt{x^2+y^2}$$ $\textsf{and bounded above by the sphere}$ $$\displaystyle x^2+y^2+z^2=128$$ $\textsf{ using triple integrals}$ \begin{align*}\displaystyle V&=\iiint\limits_{R}p(x,y,z) \, dV...

\begin{align}\displaystyle v_{\tiny{s6.15.6.3}}&=\displaystyle \int_{0}^{1}\int_{0}^{z}\int_{0}^{x+z} 6xz \quad \, dy \, dx\, dz \end{align} $\text{ok i kinda got ? with $x+z$ to do the first step?}\\$ $\text{didn't see an example}$

$\tiny{244 .15.7.24}$ $\textsf{Evaluate the spherical coordinate integral}\\ \begin{align}\displaystyle DV_{24}&=\int_{0}^{3\pi/4} \int_{0}^{\pi} \int_{0}^{1} \, 5\rho^3 \sin^3 \phi \, d\rho \, d\phi \, d\theta \\ &=\int_{4}^{3\pi/4}...

$tiny{244.T.15.5.11}$ $\textsf{Evaluate the triple integral}\\$ \begin{align*}\displaystyle I_{\tiny{11}}&=\int_{0}^{\pi/6}\int_{0}^{1}\int_{-2}^{3} y\sin{z} \, d\textbf{x} \, d\textbf{y} \, d\textbf{z}\\ &=\int_{0}^{\pi/6}\int_{0}^{1}...

\begin{align*}\displaystyle I_{15.5.8}&=\int_{0}^{\sqrt{2}} \int_{0}^{3y} \int_{x^2+3y^2}^{8-x^2-y^2} dz \ dy \ dx \\ &=\int_{0}^{\sqrt{2}} \int_{0}^{3y} \Biggr|z\Biggr|_{x^2+3y^2}^{8-x^2-y^2}\\ &=\int_{0}^{\sqrt{2}} \int_{0}^{3y} 8-2x^2-4y^2 \ dy \ dx \\ &=\int_{0}^{\sqrt{2}}\Biggr|8y-2x^2...

Homework Statement Evaluate the triple integral y^2z^2dv. Where E is bounded by the paraboloid x=1-y^2-z^2 and the place x=0. Homework Equations x=r^2cos(theta) y=r^2sin(theta) The Attempt at a Solution I understand how to find these three limits, -1 to 1 , -sqrt(1-y^2) to sqrt(1-y^2) , 0 to...

Hey! :o Let $D$ be the space $\{x,y,z)\mid z\geq 0, x^2+y^2\leq 1, x^2+y^2+z^2\leq 4\}$. I want to calculate the integral $\iiint_D x^2\,dx\,dy\,dz$. I have done the following: We have that $x^2+y^2+z^2\leq 4\Rightarrow z^2\leq 4-x^2-y^2 \Rightarrow -\sqrt{4-x^2-y^2}\leq z\leq...

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 The first part of the question was to describe E the region within the sphere ##x^2 + y^2 + z^2 = 16## and above the paraboloid ##z=\frac{1}{6} (x^2+y^2)## using the three different coordinate systems. For cartesian, I found ##4* \int_{0}^{\sqrt{12}} \int_{0}^{12-x^2}...

Write interated integrals in spherical coordinates for the following region in the orders $dp \, d\theta \, d\phi$ and $d\theta \, dp \, d\phi$ Sketch the region of integration. Assume that $f$ is continuous on the region \begin{align*}\displaystyle...

So My Question Is Simple, But It confuse me too much! What Is Difference between the notation ∫d3x and ∫∫∫dxdydz ?

So when finding the Area from a double integral; or Volume from a triple integral: If the curve/surface has a negative region: (for areas, under the x axis), (for volumes, below z = 0 where z is negative) What circumstances allow the negative regions to be taken into account as positive when...

I recently came across a problem in Irodov which dealt with the gravitational field strength of a sphere. Took some time to get my head around it and figure how to frame a triple integral, but it felt good at the end. Am I going to start seeing triple integrals in the freshman year tho? If so...

Evaluate the triple integral. Let S S S = triple integral The function given is 2ze^(-x^2) We are integrating over dydxdz. Bounds pertaining to dy: 0 to x Bounds pertaining to dx: 0 to 1 Bounds pertaining to dz: 1 to 4 S S S 2ze^(-x^2) dydxdz S S 2yze^(-x^2) from y = 0 to y = x dxdz S...

Use a triple integral to find the volume of the solid bounded by the graphs of the equations. x = 4 - y^2, z = 0, z = x I need help setting up the triple integral for the volume. I will do the rest.

Evaluate the integral \iiint\limits_{ydV}, where V is the solid lying below the plane x+y+z =8 and above the region in the x-y plane bounded by the curves y=1, x=0 and x=\sqrt{y}.

Homework Statement $$f(x,y,z)=y$$ ; W is the region bounded by the plane ##x+y+z=2##, the cylinder ##x^2 +z^2=1##, and ##y=0##. Homework EquationsThe Attempt at a Solution Since there is a plane of ##y=0##, I decided that my inner integral will be ##y=0## and ##y=2-x-z##. But after this I have...

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}...

Hey guys I've been working some triple integration problems and I've stumbled across a question that I'm having problems with So from the picture below my solution is incorrect and I can't seem to figure out where I went wrong. Is my setup for the integrals correct or is that where I've made my...

Homework Statement Homework Equations Fubini's theorum The Attempt at a Solution I drawn the diagram with the limits (for x, y, and z) and come up with something with 4 faces, 5 corners, 8 edges is that something you guys got? Thanks

Hello everyone, I have the this inquiry: if I compute de following integral: http://micurso.orgfree.com/Picture1.jpg by numerical methods I get cero as a result. I used Maxima and Mathematica and their functions for numerical integration give me an answer equal to cero. But, if I apply...

1. Homework Statement I am trying to solve a triple integral using cylindrical coordinates. This is what I have to far . But I think I have choosen the limits wrong. Homework EquationsThe Attempt at a Solution [/B]

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 why x is p(cosθ)(sinφ) ? and y=p(sinθ)(cosφ)? z=p(cosφ) As we can see, φ is not the angle between p and z ... Homework EquationsThe Attempt at a Solution

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...

In the image attached to this post, there is an equation on the top line and one on the bottom line. In the proof this image was taken from, they say this is a consequence of divergence theorem but I'm not quite understanding how it is. If anyone could explicitly explain the process to go from...

Homework Statement Let ##T \subset R^3## be a set delimited by the coordinate planes and the surfaces ##y = \sqrt{x}## and ##z = 1-y## in the first octant. Write the intgeral \iiint_T f(x,y,z)dV as iterated integrals in at least 3 different ways. Homework Equations \iiint_T f(x,y,z)dV =...

Homework Statement A circular pond with radius 1 metre and a maximum depth of 1 metre has the shape of a paraboloid, so that its depth z is z = x 2 + y 2 − 1. What is the total volume of the pond? How does this compare with the case where the pond has the same radius but has the shape of a...

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...

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 rewrite using the order dx dy dz \int_0^2 \int_{2x}^4\int_0^{sqrt(y^2-4x^2)}dz dy dx The Attempt at a Solution I am having trouble because i don't know what the full 3 dimensional region looks like but the part on the xy plane is a triangle bounded by x = 0 , y = 4 and y =...