What is Surface area: Definition and 450 Discussions
The surface area of a solid object is a measure of the total area that the surface of the object occupies. The mathematical definition of surface area in the presence of curved surfaces is considerably more involved than the definition of arc length of one-dimensional curves, or of the surface area for polyhedra (i.e., objects with flat polygonal faces), for which the surface area is the sum of the areas of its faces. Smooth surfaces, such as a sphere, are assigned surface area using their representation as parametric surfaces. This definition of surface area is based on methods of infinitesimal calculus and involves partial derivatives and double integration.
A general definition of surface area was sought by Henri Lebesgue and Hermann Minkowski at the turn of the twentieth century. Their work led to the development of geometric measure theory, which studies various notions of surface area for irregular objects of any dimension. An important example is the Minkowski content of a surface.
Homework Statement
http://i47.tinypic.com/1z6naa.jpg
Note... I used wolfram alpha to get the answer, I did not get it myself... So I still need help. The answer shown is correct, so you'll know if you got it.
Homework Equations
Integral [0, ln(4)] sqrt(1+(dy/dx)^2)
The Attempt at...
Consider the region R bounded by f(x)=x^3, y =1, and x = 2. Find the surface area when f(x) is revolved around the line x=5.
I know that the equation for surface area is the intergal from a to b of 2pi*r*h dx
in all of the examples that I had done in class, everything was around the x or y...
Sorry I am new to the forum, I don't know how to type in the integrals and stuff.
Homework Statement
Let S be the portion of the surface y = x2 where 0 <= z <= X <= 2. Compute the surface area of S.
Homework Equations
r(u,v) = x(u,v)i + y(u,v)j + z(u,v)k
A(S) = integral ru X rv dAThe Attempt...
Homework Statement
Find the area of the surface obtained by rotating the curve y=1+5x2 from x=0 to x=8 about the y-axis
Homework Equations
As=\int2\pig(x)sqrt(1+[g'(x)]2)
The Attempt at a Solution
I changed the limits to suit the y direction, the lower limit becomes 1 and the...
Homework Statement
A tungsten filament in a lamp is heated to a temperature of 2.30 x 10^3 K by an electric current. The tungsten has an emissivity of 0.31. What is the surface area of the filament if the lamp delivers 35.0 W of power?
Homework Equations Stefan's Law
(a greek letter...
My professor told me that the neutrons in a reactor see the surface area of nuclei, and not the the projected cross section. That really doesn't make intuitive sense. Is he right?
Homework Statement
How do you solve the surface area of a sphere using Riemann Sums?Homework Equations
The Attempt at a Solution
I started out with
2 * (lim n->∞ [ (i=1 to n) ∑ [ 2*pi*(√(r^2 - (i/rn)^2))*(r/n) ] ])
where the summation is the surface area of the cylinders (or discs) inside a...
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 2\pi*f(x) dx
wouldn't this add up the differential...
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...
1. Homework Statement [/b]
For the surface with parametric equations r(s,t) = <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 <= 1.
2. Homework Equations [/b]
3. The Attempt at a Solution [/b]
I already...
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?
Hi All,
I have a problem in which I am trying to calculate the force of the windspeed, x, on a surface area of a moving object, y.
I am applying this to a monorail model, where a monorail carriage (lets for this purpose assume its a rectangular box) with a surface area on one side is y, is...
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
Hi, I have to do a project in Fortran based on solving a system. My professor mentioned one idea to me, I am trying to see if this idea is even feasible and some potential ways to progress through it before I submit a project proposal to do it.
The problem is based on...
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...
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...