Surface area Definition and 440 Threads

a derivation of the formula for arc length is simple enough: given a function f[x], find the length of the arc from x0 to x1. lim(x1-x0)/n=dx n->inf x1 S=^{i=n-1}_{i=0}\sum\sqrt{(x+(i+1)dx-(x+idx))^2+f(x+(i+1)dx)-f(x+dx))^2} xo...

I know that the surface area of a revolution is equal to the integral from a to b of 2pi times the radius time the arc length. But, why isn't it just the integral from a to b of the circumference?

This question is about the surface area of a function defined as f(x) rotated around the x-axis Now I understand how the ACTUAL integral works to find the surface area, but I'm wondering why a different integral wouldn't work... \int [SIZE="3"]2\pi*f(x) dx wouldn't this add up the...

Let's say I have a four dimensional cube. Would it have a true surface area? I'm wondering if maybe it would have a surface volume rather than a surface area.

I stumbled across a problem in one of my old math books, and the answer key is either wrong, or there's something I'm missing. Homework Statement All units are in inches. Find the approx. surface area (in inches squared) of this object: A cylinder with a right circular cone on top of it...

Surface area double integration...interval problem.. Find the area of the portion of the cone z=\sqrt{x^2+y^2} that lies inside the cylinder x2+y2=2x how to determine the interval of the double integration?

I want to know how Surface areas enclosed by Cylinders , Cones..etc can be calculated using calculus integration...

How would I find the area of the surface z=2-2x^(3/2) in the first octant and to the left of the plane x+y=1? Could someone solve this and explain to me how to do it, because I really am unsure of what to do. Thanks in advance.

Homework Statement I need help proving how you could use evaluation of the surface integral \oint\oint f(x,y,z)dS to show that the surface area of the upper hemisphere of radius a is 2\pi a2. So any ideas? Homework Equations The teacher mentioned that the divergence theorem would be...

Hi, I'm a biology PhD student looking for some help on how to calculate (or estimate) the surface area of an ellipsoid truncated parallel to the long axis. Any help would be greatly appreciated. Thanks, Murphy24

Homework Statement Find the surface area of that part of the plane 8x+3y+z=9 that lies inside the elliptic cylinder (x^2/64) + (y^2/9) =1 Homework Equations not sure what equations i need to use. probably parametrization of a region The Attempt at a Solution i'm not quite sure...

Homework Statement What's the surface area of the following 3D curve over the restricted range: z=f(x,y)=\sqrt{x^2+y^2} 0\leqf(x,y)\leq8 Homework Equations **The answer is \sqrt{2}\pi** The surface area equation (with partials) \sqrt{1+(Fx)^2+(Fy)^2} Reduces to \sqrt{2} So...

Surface area of a cone--inconsistency? Geometry tells us that the surface area of a cone with a circular base is SA = \pi rs where s is the slant height of the cone, or SA = \pi r \sqrt{r^2 + h^2} Take a cone with a circular base of radius 1 and a height of 4. This formula tells us...

I am trying to measure a concrete structure to compute the surface area. I have included a sketch of the structure with the dimensions that I have measured. This is part of a drainage canal bank where two canals intersect. The banks along both canals are paved with concrete. I tried to...

Homework Statement two metal plates 10mm thin are held a distance 40mm apart with a P.D. 10V across them. +ve plate on left. In between the plates is a vacuum (free space basically). Calculate the charges and electric fields within this system.Homework Equations Q=VC C = εo.A/d No surface area...

Homework Statement The dimensions of a closed rectangular box are measured as 80cm, 60cm, and 50cm,respectively, with a possible error of 0.2cm in each dimension. Use differentials to estimate the maximum error in calculating the surface area of the box The Attempt at a Solution I...

Homework Statement A solid cylinder with a radius of 5.0 cm and a length of 10. cm is cut in half lengthwise. What is the surface area of one of the two resulting objects? Homework Equations SA = 2(pi)(r2) + L(pi)(r2) The Attempt at a Solution Is that the correct formula? If it...

1. Find the surface area generated by rotating the curve about the y-axis 2. X=(1/3)Y^(3/2)-Y^(1/2) between 1 and 3 3. x' = (1/2)Y^(1/2) - (1/2)Y^(-1/2) x'^2 = (y^2-2y+1)/4y I'm not sure if that derivative is correct or if implicit differentiation is required. I also can't get any...

Hi! I was reading about the relationship between the surface area of Earth that reflects light and the one that radiates. As the whole Earth emits radiation, the area that would radiate is A=4\Pir2, where r is the radius of earth. But the weird part is that the textbook says that only one...

Homework Statement I need the ratio like in the image below http://img228.imageshack.us/img228/4015/foiessa.jpg I guess as a function of D. Homework Equations Volume of a cylinder: pi*r^2*h Area: 2*pi*r*(h+r)The Attempt at a Solution No clue.

1. I am suppose to find the surface area of the curve y=sqrt(4-x^2) from -1 to 1 when it is revolving around the x-axis. 2. Homework Equations : S= 2PIf(x)sqrt(1+(dy/dx)^2)dx 3. I found the derivative to be -x(4-x^2)^-1/2 and then squared it so the problem is 2Pi -1\int1...

Surface area by revolving a curve problem, help! can someone show me how to solve for the surface area when rotating y=sqrt(4-x^2) around the x-axis from -1 < x < 1. shows the steps please! thanks

Homework Statement Given that a solid cylinder has a fixed volume V, prove that its total surface area S is minimum when its height and base diameter are equal. Homework Equations derivative The Attempt at a Solution I am able to prove that question. V=\pi r^2 h...

Would it be possible to calculate the time it would take for heat to reach a given temperature between two rubbing surfaces and would the size/ area of contact influence the time if the co-efficient of friction and applied force remains constant. E.g:Let's say you have two wheel brake assemblies...

I assume there is something wrong with my thinking, but couldn't you be able to speed up the decay of radioactive waste by spreading it over a large surface area so that it could disperse its energy much better? I understand half life refers to the majority bulk of a material, but if you were to...

Homework Statement For the surface with parametric equations r(st)=<st, s+t, s-t>, find the equation of the tangent plane at (2,3,1). Find the surface area under the restriction s^2 + t^2 <=(lessthanorequalto) 1 Homework Equations The Attempt at a Solution I already figured out...

Hi guys. What is [SIZE="5"]surface area of N dimensional ellipsoid? Any help is really appreciated.

Homework Statement Find the surface area traced out when the curve 8y^2 = x^2(1-x^2) is revolved around the x-axis. The Attempt at a Solution x-axis means y = 0 When y = 0, x = 0, -1 or 1. Since this curve is "the infinity symbol", the curve has symmetry at x = 0. Isolating y, 8y^2 =...

Homework Statement A closed cylinder has total surface area equal to 600\pi . Show that the volume, Vcm3, of this cylinder is given by the formula v = 300\pi-\pi r^3 , where r cm is the radius of the cylinder. Find the maximum volume of such a cylinder. Homework Equations...

Homework Statement The question asks, "Find the surface area of the cap cut from the sphere x^2+y^2+z^2=2 by the cone z = sqrt(x^2+y^2)" The answer should be 2pi(2-sqrt(2)) My main problem is not knowing how to get started. Homework Equations With the example given, it seems we need...

Hello people. I've got an action which needs minimising \int dr \ r \sqrt{U'^{2}+U^{4}} Where U(r). Simply plugging this into the EL equations yields a nasty looking 2nd order nonlinear differential equation. I'm just wondering if there's an easier way of solving for U(r). I've tried...

Homework Statement Calculate the area for 3D sphere. Homework Equations I know there's this formula for surface of revolution: A=2\pi\int_{a}^{b}f(x)\sqrt{1+ f'(x)^2}\:\mathrm{d}x The Attempt at a Solution I thought of dividing the the sphere into slices, each of which contains a...

Homework Statement Consider now a shape that is obtained by revolving the “infinitely long” function f (x) = 1/x , 1 ≤ x < ∞ around the x-axis. Find both the surface area of the resulting object, and the enclosed volume of it, i.e. the volume of the solid obtained from revolving the...

4 \pi r^2 = SA for a sphere. everybody knows that. but, how do you derive this by integrating infinitesimal amounts of area over the curvature of the sphere?

Homework Statement I apologize for the mass questions. I am very most confused with this one surface area generated by revolving around y-axis the curve y=cuberoot(x) from y= 1 to 2. Homework Equations S 2pi*g(y) \sqrt{1-(g'(y))^2} The Attempt at a Solution i found that g(y)=y3...

Homework Statement Given the elliptic paraboloid of height H and two semiaxes A and B. How to find its surface area? Homework Equations x = A * sqrt(u) * cos(v) y = B * sqrt(u) * sin(v) z = u u belongs to [0; H], v belongs to [0; 2*PI)

When calculating the volume of a sphere, what does (4/3) represent? Why is it (4/3) * pi * r^3 .. and not some other number/fraction? I'm also curious about the surface area of equilateral triangle. Why is it sqrt(3)/4 * a^2 ... What does sqrt(3)/4 physically represent in the geometry?

Basically I have a horizontal tank with elliptical heads on each end. Given any liquid height, I would like to calculate the surface area which is in contact with the liquid. The heads can be represented by a general oblate spheroid described by equation (x2 + y2)/a2 + z2/c2 = 1. I can...

Hello, Can you help me in this question , In part a I must find the volume of cup(cylinder) In part b I must find radius but how can I find Area In part c ,d I don't know how can I solve it?? What is spanning angle and surface area?? Note : this question from previous exam but I...

Greetings, I'm trying to find the surface area of the part of the sphere x^{2}+y^{2} + z^{2}=1 above the cone z=\sqrt{x^{2}+y^{2}}. I know, that a surface area of a surface r(u,v) = x(u,v) + y(u,v) + z(u,v) can be given by, A(S) = \int\int | r_{u} \times r_{v} | dA A function, z=f(x,y) can...

A silo has cylindrical wall, a flat circular floor, and a hemispherical top. If the cost of construction per square foot is twice as great for the hemispherical top as for the walls and the floor, find the ratio of the total height to the diameter of the base that minimizes the total cost of...

i need MAJOR HELP! this is the problem: The can do tin can company minimizes costs by construckting cans from the least possible amount of material. this company suppplies many different sizes of cans to packing firms. a designer with the company needs to determine the radius that gives...

Homework Statement The curve is rotated about the y axis, find the area of the resulting surface. y=(1/4)X2-.5ln|x| 1<_X<_2 Homework Equations S=2(pi)(f(x))\sqrt{}1+f'(x)^2 The Attempt at a Solution Alright I'm not entirely sure where to even begin. Since I'm rotating about the Y-axis I know...

Two vessels of the same volume, in the same location and open to the atmosphere. One is tall and of a small diameter, the other is short and is of a larger diameter. Each vessel supplied with an identical heat source. Each vessel filled with an equal volume of an identical liquid. The goal...

Homework Statement A charge distributed uniformly over 1 face of the circular disk, find the surface charge density ? well i know density for surface charge is S= charge/area, the area for face of 1 circular disk is what ? 2pir ?

I have a problem that I've been stuck on for a while as follows, Find the surface area of the part of the cylinder x^{2}+y^{2}=2ay in the first octant that lies inside the sphere x^{2}+y^{2}+z^{2}=4a^{2}. Express your answer in terms of a single integral in \phi, you do not need to evaluate...

Homework Statement I have this function, and I need to find the Surface area of the function that is confined between the plains x=0, y=0 and z=0. My question is, what's D? Homework Equations The Attempt at a Solution

Homework Statement Find the area of the surface with parametric equations x=uv, y=u+v, z=u-v. u^2 + v^2 <= 1Homework Equations A(S)= double integral over the domain D of the norm of the cross product (r_u X r_v) DAThe Attempt at a Solution I started off with finding the norm of the cross...

Homework Statement I need help on an optimization problem involving a hexagonal prism with no bottom or top, but the top is covered by a trihedral pyramid which has a displacement, x, such that the surface area of the object is at a minimum for a given volume. The assigned variables include...

Homework Statement Find the area of the sphere x^2 + y^2 + (z-2)^2 = 4 that lies inside paraboloid z = x^2 + y^2Homework Equations The Attempt at a Solution when i take the equation of the spere and replace x^2 + y^2 with z i get: z(z-3) = 0 so they intersect at the plane z = 3. were supposed...