Surface area Definition and 440 Threads

How do you prove that for a given volume, sphere has the minimum surface area?

So, I have been trying to figure out how to derive the equation for the surface area of a sphere. All attempts have resulted in colossal failure and as such are not even worth posting on the forum. I know Archimedes was the first to come up with the formula but I have not been able to find...

if we want to find the volume of a function revolved about the x-axis all we do is find the differential element.. dV=(Area)dx =\piy2 , where y=y(x) so then.. \int\piy2dx the differential element looks like this... http://imgur.com/HwrihKA,hjkKFzV,cpUPsvZ,YKIyTIB so i add a bunch...

is it because that pressure in liquid acts in all directions?

A surface is obtained by rotating around the x-axis the arc over the integral(-1,0.5) of an ellipse given by: x^2+4y^2=1 What is its surface area? Here's my solution: I use the equation: S=integral( upper bound: a lower bound: b ) 2(pi)y*[1+(f'(x))^2]^0.5 dx Since x^2+4y^2=1...

θHomework Statement The attached diagram depicts a sphere with several variables: the height of the cylinder, the radius of the cylinder and an angle. All that has been given to me is the hypotenuse of a triangle used. Homework Equations To my knowledge I was told to use 2sinθcosθ=sin2θ...

Got a little problem here, just want to make sure what I'm doing is right because it's a little different from anything I've done that is using the same formula initially. So there's a sphere denoted as \(\Omega\): $$x^2+y^2+z^2=R^2$$ and its cut by two planes \(z=a\) & \(z=b\) where...

I am curious about the relationship between an ever expanding sphere's radius and its surface area. how would I relate the rate of change in radius to the rate of change in the surface area.

Homework Statement Find the surface area of the surface defined by: z = (2/3)(x^(3/2) + y^(3/2)), 0 ≤ x ≤ 1, 0 ≤ y ≤ 1 Homework Equations SA = ∫∫ √(fx)^2 + (fy)^2 + 1 dA The Attempt at a Solution Solved for the partial derivatives and plugged them into the formula, but got a...

S is a portion of a curve with r(u,v) where 0 < u < 2 and 0 < v < 2pi I'm meant to calculate Flux of the vector field F My Calculations First found dr/du then dr/dv Using the cross product, I found N = (- u cos (v) + 5 sin (v), -5 cos (v) - u sin(v), u) Then I dot product with the given...

Homework Statement x^2 + (y - 2)^2 = 1 The hint given by the question was to split the function into 2Homework Equations Surface Area about x axisThe Attempt at a Solution So i did this. (y - 2)^2 = 1 - x^2 y = √(1 - x^2) +2 and y = 2 - √(1 - x^2) The range I calculated when...

Calculate the area of ​​the surface of rotation which occurs when the curve rotate in y-axe. I start with $$x=\sqrt{28y}$$ then $$f'(x)=\frac{14}{\sqrt{28y}}$$ so we got $$2\pi\int_0^{21}\sqrt{28y}\sqrt{1+(\frac{14}{\sqrt{28y}})}^2$$ then I rewrite as...

Homework Statement (i) Find the normal, n, at a general point on the surface S1 given by; x2+y2+z = 1 and z > 0. (ii) Use n to relate the size dS of the area element at a point on the surface S1 to its projection dxdy in the xy-plane. The Attempt at a Solution To...

Homework Statement Find the surface area of the solid obtained by rotating y = √9 − X^2 , − 2 ≤ X ≤ 2 about the X-axis. Homework Equations ∏r^2 The Attempt at a Solution Y = (9 - x^2)^-1/2 Is that can do like this.. then how bout the power inside the X^2??

edit: vector* analysis; sorry for the typo Homework Statement Given that A = ||ru||2, B = ru\bulletrv, C = ||rv||2 surface area of S is Area(S) = \int^{d}_{c}\int^{b}_{a} = sqrt (AC - B2) dudv The Dirichlet energy can be thought of as a function as follows E(S) =...

Hey, folks. I'm trying to derive the surface area of a sphere using only spherical coordinates—that is, starting from spherical coordinates and ending in spherical coordinates; I don't want to convert Cartesian coordinates to spherical ones or any such thing, I want to work geometrically...

Homework Statement The given curve is rotated about the y-axis. Find the area of the resulting surface. y = 3 - x^2 0 \leq x \leq 4 Homework Equations A_{y} =2\pi\int_a^b x \sqrt{1+\left(\frac{dx}{dy}\right)^2} dy The Attempt at a Solution now we need to write x in terms...

Is it possible to come up with a derivation of the surface area of a sphere without using a double integral? Most of the ones I've found seem to involve double integrals; For example, this was given as the "simplest" explanation in a thread from 2005: S=\iint...

1. Use the integration capabilities of a graphing utility to approximate the surface area of the solid of revolution. (Round your answer to four decimal places.) Function: y = sin(x) Interval: [0, pi/4] revolved about the x-axis 2. Use the area of a surface of revolution...

Homework Statement Use Pappus' Theorem for surface area and the fact that the surface area of a sphere of radius c is 4(pi)c^2 to find the centroid of the semicircle x= sqrt ( c^2 - y^2) Homework Equations S = 2 (pi) * p * L where s=surface area; p=distance from axis of revolution...

Homework Statement Suppose we consider different flying objects, and that each object is characterized by a linear dimension l. Part A: Use dimensional arguments to show that the volume V scales with size as V \sim l^{3} and that the surface area scales as S \sim l^{2}. Part B: Show...

As far as I know, friction force is equal to the product of the normal force and coefficient of friction, hence is independent of surface area. So why is it that race cars have wider tyres than conventional vehicles?

Hi all, Thanks for response :) I Dont really understand what is surface integrals ?? and its difference with Surface Area using double integrals. Can anyone help ? thanks a lot...

I'd read that friction is independent of the surface area of the bodies in contact. But somewhere in the internet I found that this explanation was just a good approximation and that friction actually depends on area. Can anyone explain a bit more on this?

We know that we calculate the volume of sphere by taking infinitesimally small cylinders. ∫ ∏x^2dh Limits are from R→0 x is the radius of any randomly chosen circle dh is the height of the cylindrical volume. x^2 + h^2 = R^2 So we will get 4/3∏R^3 Now the question is why cannot we...

Homework Statement Oil is released from a submerged source at a constant flow of rate of Q=(0.1 m3)/s. Density of oil (p=870 kg/m3) is less than that of water and since oil is only sparingly soluble in water a slick will form on the water surface. Once its formed it will have a tendency to...

Homework Statement Calculate ∫∫f(x,y,z)DS for the given surface and function. Part of the plain x+y+z=0, contained in the cylinder x^2+y^2=1 f(x,y,z)=z^2. Homework Equations ∫∫F(x,y,z)Ds=∫∫F(g(u,v)*||n(u,v)|| N= TuXTv Tu= G(u,v)(du); Tv is the same only the derivative is with respect to v...

1. Homework Statement I need some help with a surface area of a solid. The solid is made from rotating the line y=x^2 around the x axis. So it's sort of like a cone or a horn. Here are my steps: 2. Homework Equations Surface of revolution formula Integrate 2∏r times the square root of 1...

So if we are looking to find the surface area of a solid of revolution formed by rotating a curve about a line, we can use the following: S = ∫2(pi)yds (if rotating about the x-axis) FOr example, say our curve is y=x2 and we want to find the surface area of the solid of revolution...

Homework Statement Consider the region of the x y plane given by the inequality: x^2 + 4x + y^2 - 4x - 8 ≤ 0; If this region rotates an angle of π/6 radians around the line given by the equation x + y = 0, it will create a solid of revolution with surface area equal to (i)...

Ok, so if I have a square that is exactly 10 inches by 10 inches, then the surface area is 100 square inches exactly. But if I roll up that square into a tube and calculate its surface area, it's 2∏r times the length. And since the calculation involves ∏, that means I won't get an exact answer...

I want to calculate the surface area of semi cylinder with one or both ends skew at a given angle. To calculate the area/perimeter i need to find out the end profile of the semi cylinder. If the skew plane is perpendicular to the cylinder axis, the cut surface would be half ellipse with major...

Homework Statement Calculate the area of a circle of radius r (distance from center to circumference) in the two-dimensional geometry that is the surface of a sphere of radius a. Show that this reduces to πr2 when r << a Homework Equations Surface area of a spherical cap = 2πah = π(r2 +...

You are pushing a wooden crate across the floor at a constant speed. You decide to turn the crate on end, reducing by half the surface area in contact with the floor. In new orientation, to push the same crate across the same floor with the same speed, the force that you apply must be about: A)...

Ok, so I did a test today for advance functions and there was a question: the volume of a pop can is 350ml, find the minimum surface area and determine the dimensions. Where I got stuck:350=pi(r)^2*h h=350/pi(r)^2SA= 2pi(r)^2+2pi(r)( h) SA=...

Ok, so I did a test today for advance functions and there was a question: the volume of a pop can is 350ml, find the minimum surface area and determine the dimensions. Where I got stuck: 350=pi(r)^2*h h=350/pi(r)^2 SA= 2pi(r)^2+2pi(r)(h) SA= 2pi(r)^2+2pi(r)(350/pi(r)^2)...

Homework Statement Grains of fine California beach sand are approximately spheres with an average radius of 50 μm and are made of silicon dioxide, which has a density of 2.6 × 103 kg/m3. What mass of sand grains would have a total surface area (the total area of all the individual spheres)...

So the first question is to find the surface area of a torus generated by rotating the circle (or shall I say semi circle) y=√r-x2 around y=r if the idea is to find the surface are of the half torus and then multiply by 2, wouldn't it be the same for the circle (or shall I say semi circle...

I met a proof problem that is as follows. ##\bf a = ∫_S d \bf a##, where S is the surface and ##\bf a ##is the vector area of it. Please proof that ##\bf a = \frac{1}2\oint \! \bf r \times d\bf l##, where integration is around the boundary line. Any help would be very appreciated!

Homework Statement Find the area of the surface obtained by rotating the curve y=x3, 0≤x≤2 about the x-axis. Homework Equations \begin{equation*} SA = \int_{0}^{2} 2 \pi y L \end{equation*} The Attempt at a Solution SORRY, I don't know how to use LaTeX yet. ∫2∏y√(1+(dy/dx)2)dx from 0->2...

Hello, I have actually asked a similar question before, but I just realized something and I want to edit the question now: I am trying to derive the formula for the lateral surface area of a cone by cutting the cone into disks with infinitesimal height, and then adding up the lateral areas...

Hello, this is my first time posting on physics forums, so if I do something wrong, please bear with me :) I am trying to derive the formula for the lateral surface area of a cone by cutting the cone into disks with differential height, and then adding up the lateral areas of all of the...

Homework Statement Having recently learned the disk/shell/washer method for finding the volume of a solid of revolution, I'm trying to apply similar methods to derive the formula for the surface area of a cone (and hopefully after that, that of a sphere). The region that is revolved around...

For a given function f(x)=x, if we rotate this function from [0,1] around the x-axis, we'll have a cone... why can't I find its surface area by adding the perimeter of all the circunferences from [0,1] with radius = f(x)?Something like 2*pi∫xdx? The formulas says we have to do 2*pi∫x*√(i+y'²)dx ...

Dimensions -- Disc Brake Surface Area During frequent braking under race conditions the disk brake rotors on the car described above reach a temperature of 500C. These disk brakes rely on forced convection to cool them. The dimensions of each disk rotor are: outer radius 130mm; inner radius...

[SIZE="5"]how the rate of heterogeneous reaction changes with the surface area of catalyst. can anyone tell me about the equation relating both quantities?

I don't know how to calculate the surface area after setting everything up. I have tried both MAPLE 15 program and wolfram alpha, but I can't find the answer. I have the function: f(x,y)= x*(y^2)*e^-((x^2+y^2)/4) The form I found for surface are was the square root of (sum of squares of...

Homework Statement Find the surface area of the portion of the cylinder x^2+y^2=4y lying inside the sphere x^2+y^2+z^2=16Homework Equations Double integral for finding surface area. \iint_S d\sigma The Attempt at a Solution At first, i thought it was just the area of the surface of sphere...

Homework Statement A small sphere of mass m carries a charge of q. It hangs from a silk thread which makes an angle θ with a large charged non-conducting sheet. Calculate the surface charge density on the sheet. Homework Equations ∫E.ds The Attempt at a Solution I found E of...

Homework Statement Find the area of the surface z=7+2x+2y2 that lies above the triangle with vertices (0,0) (0,1) and (2,1) The Attempt at a Solution It's not a particularly difficult problem to set up, I just can't seem to get a simple integrand. ∫∫√(16y2+5) dA is what I come up...