Area is the quantity that expresses the extent of a two-dimensional region, shape, or planar lamina, in the plane. Surface area is its analog on the two-dimensional surface of a three-dimensional object. Area can be understood as the amount of material with a given thickness that would be necessary to fashion a model of the shape, or the amount of paint necessary to cover the surface with a single coat. It is the two-dimensional analog of the length of a curve (a one-dimensional concept) or the volume of a solid (a three-dimensional concept).
The area of a shape can be measured by comparing the shape to squares of a fixed size. In the International System of Units (SI), the standard unit of area is the square metre (written as m2), which is the area of a square whose sides are one metre long. A shape with an area of three square metres would have the same area as three such squares. In mathematics, the unit square is defined to have area one, and the area of any other shape or surface is a dimensionless real number.
There are several well-known formulas for the areas of simple shapes such as triangles, rectangles, and circles. Using these formulas, the area of any polygon can be found by dividing the polygon into triangles. For shapes with curved boundary, calculus is usually required to compute the area. Indeed, the problem of determining the area of plane figures was a major motivation for the historical development of calculus.For a solid shape such as a sphere, cone, or cylinder, the area of its boundary surface is called the surface area. Formulas for the surface areas of simple shapes were computed by the ancient Greeks, but computing the surface area of a more complicated shape usually requires multivariable calculus.
Area plays an important role in modern mathematics. In addition to its obvious importance in geometry and calculus, area is related to the definition of determinants in linear algebra, and is a basic property of surfaces in differential geometry. In analysis, the area of a subset of the plane is defined using Lebesgue measure, though not every subset is measurable. In general, area in higher mathematics is seen as a special case of volume for two-dimensional regions.Area can be defined through the use of axioms, defining it as a function of a collection of certain plane figures to the set of real numbers. It can be proved that such a function exists.
Homework Statement
A rectangle has one vertex in quadrant I at the point (x,y) which lies on the graph of y = 2x^2 and another vertex at the point (-x, y) in the second quadrant and the other vertices on the x-axis at (-x, 0) and (x, 0)
What is the domain of the area function?
y = 2x^2 = w
l...
Data
From the rectangular glass sheet ABCD the isosceles triangular part ADE is cut away (See figure)
The length of CE is 1m.
Problem
i. Take the length of DE as x meters, write an expression in terms of x , for the area of the remaining part of the sheet.
The area of the remaining part...
Find the area bound by the curve y=x^2-16 , the x-axis and the lines x=2 and x=5. i am trying to use the definite integral way ∫_2^5. but i am not getting the right answer. the right answer is 53/3
Homework Statement
How large a surface area in units of square feet will 1 gallon of paint cover if we apply a coat of paint that is 0.1cm thick?
Homework Equations
Since Volume is L * W * H and we can assume the object is square besides the height which in this case will be the thickness. So...
Homework Statement
The three vectors A, B, and C point from the origin O to the three corners of a triangle. Show that the area of the triangle is given by 1/2|(BxC)+(CxA)+(AxB)|.
Homework EquationsThe Attempt at a Solution
I know that the magnitude of the cross product of any two vectors...
1. Homework Statement
calculate the area and perimeter of 2 rectangles(two objetcs and one builder), print the sides, area and perimeter, the function printrectangle must identify which side belongs to base and height...
the teacher suggest this in private: float side1 float side2
and this in...
https://www.physicsforums.com/threads/mechanics-of-materials-homework.855915/']<Moderator's[/PLAIN] note: thread moved from a technical forum, so homework template missing.>
https://www.physicsforums.com/threads/mechanics-of-materials-homework.855915/
This is a link to a problem which I am...
Hey everyone,
I've been stuck on this one piece of HW for days and was hoping someone could help me.
It reads:
The surface area, A, of a sphere with radius R is given by
A=4πR^2
Re-derive this formula and write down the 3 essential steps. This formula is usually derived from a double...
Hi everyone!
Please help, I have spent some considerable time to understand the two concepts and still this is nagging at me... I am relating to Structural Engineering, just to let you know guys. My question is ..
Moment of inertia is about distribution of mass, the further away from the axis...
Homework Statement
For the relationship ##|y| = cos(x-y), -\frac {π} {2} ≤ x ≤ \frac {π} {2}##, use calculus and algebra to determine the total area enclosed between the graph and and the x-axis.
Homework Equations
Area between points ##a## and ##b## = ##∫_a^b f(x)dx## given that ##f(x)>0##...
I like science a lot. Research, discovering new things, math, and anything to do with space is exciting to me. I also like the application of physics, like engineering. I was thinking about majoring in space physics, astronomy, experimental, applied, or engineering physics. Maybe a minor...
I'm given this problem to solve with the assumption as stated above. The answers need not be logical as long as the linkage mechanism can be simulated. I've attempted the question using Mercedes Mono Wiper mechanism but only manage to cover 41% given the width of the wiper. May I ask if there...
Homework Statement
I have a confusion regarding areas.
Usually in fluid flow I am using the A=(pi*D^2)/4
However in heat transfer we usually use A=pi*D*L
Could you please explain this?I mean in first case(fluid flow) it is cross sectional area and in the 2nd case(heat transfer) is like the...
Homework Statement
What is the area, and approximate uncertainty in a circle with radius 3.1*10^4 cm (or written: 3.1e4 cm)?
Homework Equations
Area=Pi*r^2
The Attempt at a Solution
My attempt to the solution took some trial and error, and it went as follows:
Substitute the circle's radius...
Homework Statement
Homework Equations
Formula for Area of a retangle : A = L x W
Pythagorean theorem: A2 + b2 = c2
The Attempt at a Solution
So I am pretty sure I did it correct but I just want to be 100% certain I will get this right, By the way its a picture cause I found it easier to...
So, i was studying some fin design in a heat transfer course , and then came the part where the efficiency is to be calculated, then i noticed that when he calculated the surface area and the sides of a rectangular fin weren't included, so i searched and i found out that it was neglected...
Homework Statement
FInd the area bounded by x=-3, y=-x^2-2x, and y=x^2-4. (Hint: Graph the picture)
2. The attempt at a solution
My professor did set up the problem in class, but its throwing me off. He set it up as the lower bound -3 to 2, with the function (2x^2+2x-4)dx. I tried solving this...
Currently revising for my A-Level maths (UK), there is unfortunately no key in the book;
Given the triangle with sides a,b,c respectively and the area S, show that ab+bc+ca => 4*sqrt(3)*S
I have tried using the Ravi transformation without luck, any takers?
AB = (x + 3) cm, DC = (2x − 3) cm and BE = EC.
area of the trapezium is 15 cm^2 \therefore,(x + 3) (2x − 3) or ? i think you should find the are and use the squarootcan you help me to proceed.
Homework Statement
i'm having problem of understanding the formula of area of moment of uniformly distributed load and uniformly varying load... the shape of graph for moment of uniformly distributed load and uniformly varying load.are similar,right...
I am reading the book "Super-radiance Multiatomic Coherent Emission" by Benedict et al. and on pg. 32, they discuss the initial conditions for a particular case covered by Burnham and Chiao (1969). It mentions that the system was "excited to a state with angle ##\theta_0## by a short coherent...
Could anyone please help me with the area of heat sink required if I want to dump heat 6 feet below the surface?
The heat to be rejected is 20000 kW
Temperature of the fluid has to be dropped from 30 deg C to 19 deg C.
I need rough estimates of the area required to lay down looped pipelines to...
Consider the following map ##f## that maps an annulus to a larger annulus: ##f: (r, \theta)\to(r+1, \theta)##. ##f## maps the annulus in the region ##1\leq r\leq2## to the annulus in the region ##2\leq r\leq3##. Clearly, the area is not preserved.
Next, is the converse true? That is, must an...
suppose I have a sphere that has radius with the vlaue of 1.
the integral is:∫∫n⊗ndA
where dA = sinθdθdφ
what is n⊗n?
I'm supposed to the area of the sphere.
This question was in an exam in continuum mechanics in the Technion
Homework Statement
A cylindrical glass tube (linear thermal expansion coefficient ##\alpha##) contains liquid (volume thermal expansion coefficient ##\beta##). The height of the tube is ##h_{t,0}## and the height of the liquid inside of it is ##h_{l,0}##. If the temperature changes of an amount...
Greetings All,
I have a rather odd question which has been bothering me. If you have a perfectly round sphere sitting on a perfectly flat plane, what is the area of surface contact between the two? Is there an actual value, or is it something which can't be calculated. I'm assuming the diameter...
Hello everyone. I have what is probably a relatively simple question. I'm trying to calculate the resistance between two rectangular copper plates submerged in water. I found this thread that briefly discusses it...
Homework Statement
can someone explain about the area of moment diagram ? taking the circled part as example , why it's 0.5(2)(800)(4/3) ?why shouldn't it be (800)(4/3) ??
Homework Statement
An object is moving around the unit circle with parametric equations x(t)=cos(t), y(t)=sin(t), so it's location at time t is P(t)=(cos(t),sin(t)) . Assume 0 < t < π/2. At a given time t, the tangent line to the unit circle at the position P(t) will determine a right triangle...
The following derives the relation that for a blackbody radiation the energy density is proportional to the energy emitted per unit area over unit time.
The average energy density ##d\psi## is obtained by dividing the radiant energy ##dE## received by the surface ##dB## in 1 second by the...
Dear All Gravitinos,
It seems that the current string theory and loop gravity's explanation for the micro-states of black
holes are all quantum mechanical and have no classic correspondence. I, in this day's arxiv, post a
pure classic interpretation for this question, titled "Black...
Homework Statement
For a gas of N fermions with mass M in 2D in a region of area A in thermal equilibrium at temperature T, we are asked to find ##U/N## in fuction of ##T## and ##a=A/N##.
The attempt at a solution
I used ##U=\sum(<n_i>\epsilon_i) = \sum(\exp(\beta(\mu-\epsilon_i))\epsilon_i...
Homework Statement
f(x) = √(x+2), g(x) = d/dx (f(x))^(f(x)). Find the total area enclosed between g(x) and √(x^2) correct to 3 decimal places.
Homework Equations
Knowledge of differentiation and integration - specifically areas between curves.
The Attempt at a Solution
I've attempted to...
I understand the theory behind this type of question well enough; you solve ln(x)=sin^2(2x)-cos(3x)+1 to find the x values at the points of intersection, and then set up definite integrals over the domains of said x-values, subtracting whichever function is below the other for a specific domain...
If you were given f(x) = 3sin(5x), would it be possible to express the total area between f(x) and the x-axis between the origin and any given intercept? Basically, could you form a general equation for the total area for f(x) where x∈[0,a] and a is an x-intercept.
Homework Statement
Please find attachedHomework Equations [/B]
Definite integrals and area between two curvesThe Attempt at a Solution [/B]
Also find attached.
The answer in the back of the book says that part c should be 10/3, but that would mean that (8sqrt (2))/3 would need to cancel out...
I am reading the wikipedia article on the Cauchy stress-tensor. The article says that given some object, let ##P## be a point in the object and let ##S## be a plane passing through that point. Then "an element of area ##\Delta S## containing ##P##, with normal vector ##n##, the force...
Summer is almost here and people tend to spend more time outdoors. That is very pleasant and most of us enjoy the sun and warm weather. But there may also be some dangerous creatures, especially in some areas. What are the most common dangerous, venomous or parasite animals in your area?
In my...
Homework Statement
Homework Equations
A = l x w
A = hbb / 2
The Attempt at a Solution
To find the area of a rectangle you would use the formula "A = L x W". In this case A = 14 x 8 which is eual to 112, Now all i need is the area of the triangle and then i add them together and that will be...
Homework Statement
"Areas of regions Make a sketch of the region and its bounding curves. Find the area of the region."
"The region inside the curve ##r = \sqrt{cosθ}## and inside the circle ##r = \frac{\sqrt{2}}{2}##.
Homework Equations
##A = \frac{1}{2}\int_α^β(f(θ)^2-g(θ)^2)dθ##
Answer as...
Hi everyone! I wanted to pose this question to those who PF members who are current PhD students or postdocs in physics. What area of physics research are you involved with?
I tried to list out all areas of physics research that I'm familiar with, but I also created an "Other" category for any...
Homework Statement
I'm trying to figure out the surface area on a 5 sided shape where the sides can all be modeled by "lunes". The shape will end up looking like a banana peel. We are modeling the sides of the shape as lunes with varying angles on a sphere of radius 3 inches. I'm trying to...
We know, that the infinitesimal area element in Cartesian coordinate system is ##dy~dx## and in Polar coordinate system, it is ##r~dr~d\theta##. This inifinitesimal area element is calculated by measuring the area of the region bounded by the lines ##x,~x+dx, ~y,~y+dy## (for polar coordinate...
Homework Statement
2. Homework Equations
A = l x w
A = b x w / 2
The Attempt at a Solution
I calculated all the areas for the triangles and rectangles. So for the first rectangle (the long one) I did (L)(w) = A. Basically 8 x 2 = 16m2 (this is the same area for the other long rectangle)...
Homework Statement
In the picture above, line y = c intersects with parabola y = 6x-x^2 in the first quadrant.
If the gray area below line y = c and the gray area above line y=c are equal, then value of c is ...
A. 19/4
B.21/4
C.23/4
D.25/4
E. 27/4
Homework Equations
Area under parabola =...
I have a diameter of orifice is 1.025 inches and it says the Plate Area is 0.0058ft ^2 how is this answer received? Ill be using it to find orifice coefficient I don't understand why its 0.0058