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.
In my working, i have the following approach;
Using area of semi -circle and area of sector concept;
##x +z = \dfrac{9π}{2}##
##x +y = \dfrac{9π}{2}##
##z+p = \dfrac{9π}{2}##
On solving the simultaneous equations,
##⇒x=p##
then,
##x=\dfrac {9π}{4} - \left(\dfrac{1}{2} ×3 ×3\right) =...
Hello! Consider this figure
Where a = 5 cm. I need to find the Circumference and Area of this figure.
For the Circumference I tried it like this. I have these 2 shapes, in red and in green
For the green one its basically 2 times a quater of a circle so it should be $$ C_1= 2 * \frac{2\pi...
Hello everyone,
I found a good proof for the area of a circle being ##{\pi}r^2## but I was recently working on my own proof and I used a change of variables and was wondering if I did it correctly and why a change of variables seems to work.
I start with the equation of a circle ##r^2 = x^2 +...
**EDIT** Everything looked good in the preview, then I posted and saw that some stuff got cropped out along the right edge...give me some time and I'll fix it.
Hello all,
I"m trying to calculate shaded area, that is, the area bounded by the curves ##x=y^{2}-2, x=e^{y}, y=-1##, and ##y=1##...
Think of a 3D rectilinear grid made of these rectangular cells, some of the cells will intersect with the sphere. I am trying to compute each intersecting area and the total sum. Ideally the total sum of the intersecting area should be close to ##4 \pi r^2##. I have not found any literature...
Mentor note: Moved from a math technical section, so template is not present.
I was asked to calculate the area of the smaller section enclosed by the circle x²+y²-6x-8y-35=0 and the x axis. I've tried to solve it with geometry, using the x-intercepts and the centre of the circle I drew a...
TL;DR Summary: Finding the power loss in a wire of varying area - my problem is I don't know how to set up the integral
Hopefully you can see in the diagram below that the area of the wire varies linearly with length. I know the equations for resistance and power loss and I can express the...
I have the following question to solve:Use the metric:$$ds^2 = -dt^2 +dx^2 +2a^2(t)dxdy + dy^2 +dz^2$$
Test bodies are arranged in a circle on the metric at rest at ##t=0##.
The circle define as $$x^2 +y^2 \leq R^2$$
The bodies start to move on geodesic when we have $$a(0)=0$$
a. we have to...
Hi all,
While calculating the surface area for an object, I was told the below statement. However, I am not sure is this correct, please can someone help me to explain this with an example? Is the below statement always true?
The surface area % increase should be in line or less than the %...
Suppose the following integration,
##\int_3^{-1} x^2 \, dx = \frac{1}{3}(-1)^3 - \frac{1}{3}(3)^3 = -\frac{28}{3}##
However, if we have a look at the graph,
The area between ##x = 3## and ##x = -1## is above the x-axis so should be positive. Dose anybody please know why the I am getting...
Wawawawawa boggled me a little bit... but finally managed it...seeking alternative approach guys;
kindly note that what i have indicated as ##*## and a ##√## is the correct working ...
Text book answer indicates ##17.5## as answer... will re check my rounding solutions later...
My working-...
I have a several questions on the following block of codes taken from ganfetex01_aux.in:
solve
save outf="ganfetex01_$'index'.str"
extract init inf="ganfetex01_$'index'.str"
extract name="2DEG" 1e-4 * area from curve (depth, impurity="Electron Conc" material="All" mat.occno=1 x.val=0.5) \...
Hello,
I am thinking about a real-life problem: the flooding of a stream in may area of town.
A stream discharge, ##Q=A v##, represents the volume of water passing through the cross-sectional area ##A## in one second as the water moves with speed ##v##.
Let's assume that the stream has 2...
Hi PF
There goes the quote:
The Basic Area Problem
In this section we are going to consider how to find the area of the region ##R## lying under the graph ##y=f(x)## of a nonnegative-valued, continous function ##f##, above the ##x##-axis and between the vertical lines ##x=a## and ##x=b##, where...
Consider the following scenario:
Given that points ##M## and ##N## are the midpoints of their respective line segments, what would be the fastest way to determine what percentage of the squares total area is shaded purple?
I managed to determine that the purple shaded area is ##5\text{%}##...
I try to divide the area of CDGE into two areas of triangles by drawing line DE.
The ratio of area of triangles ABE and ECD = 4 : 1
The ratio of area of triangles ADG and DGE = AG : GE
The ratio of triangles ADG and AGF = DG : GF
Then I don't know what to do.
Hi, the problem statement is above. I have some questions about how to calculate the area and the direction of the magnetic field of this problem.
As the magnetic flux, my professor have defined it as Phi= integral(B dS)=(Area)e_x B= (Area_triangle + (L^2/2) *(β + α(t)))*B e_z.
How can one know...
Consider a convex shape ##S## of positive area ##A## inside the unit square. Let ##a≤1## be the supremum of all subsets of the unit square that can be obtained as disjoint union of finitely many scaled and translated copies of ##S##.
Partition the square into ##n×n## smaller squares (see...
I usually think of a sphere as the set of all points ##P_x##, that have the identical distance r to some point ##C## which is the center of the sphere. I calculate the surface area ##A## of the sphere as
$$A=4 \pi (C P_x)^2$$
However, what happens if I think of the distance between the points C...
Greetings all,
I'm new here and hope I'm asking this in the correct thread. So, the question is; where you have a vacuum created by a "flow through" liquid witin a large diameter container exerting suction force upon a smaller diameter input tube submerged in a liquid, does the surface area of...
Hi!
For this problem,
Why is the area of each ring segment dA equal to (2π)(r)(dr)?
However, according to google the area of a ring segment (Annulus) is,
Many thanks!
The implicit curve in question is ##y=\operatorname{arccoth}\left(\sec\left(x\right)+xy\right)##; a portion of the equations graph can be seen below:
In particular, I'm interested in the area bound by the curve, the ##x##-axis and the ##y##-axis. As such, we can restrict the domain to ##[0...
To find the y value of the centroid of a right triangle we do
$$\frac{\int_{0}^{h} ydA}{\int dA} = \frac{\int_{0}^{h} yxdy}{\int dA}$$
What is wrong with using
$$\int_{0}^{h} ydA = \int_{0}^{b} y*ydx$$ as the numerator value instead especially since ydx and xdy are equal and where h is height of...
If I have a triangle on a sphere with two of its angles 90 degrees each, do I conclude that it's isosceles and that the shortest distance (on the sphere) beteeen the base and the vertix of the thid angle is 1/4 the circumference of a great circle on the sphere?
This is the picture I have in...
Florida hurricanes...Oklahoma tornados...These are two areas I never want to live no matter the salary (okay, for $500,000 or more, sure...I'm there!).
I have to imagine it sucks having having your house flooded/blown down every three or so years. Not to mention your loved ones possibly dying...
Summary: A 5.0- cm -diameter cylinder floats in water. How much work must be done to push the cylinder 11 cm deeper into the water?
F =Aρgx
A 5.0- cm -diameter cylinder floats in water. How much work must be done to push the cylinder 11 cm deeper into the water?
F =Aρgx
x being the...
Hi
I am looking to find the equation that determines the minimum (and if possible maximum that might damage the electrode) voltage that starts the electrolysis process for a given area of a graphite electrode in a brine solution medium (lets say 30%) at equilibrium state.
Also how does the...
I want to open up a spectrometer to see how the inside looks like. Is it true that the CCD in spectrometer is more sensitive to dusts than normal camera CCD such that even a small speck of dust can cover the CCD pixels and the user will get holes in the spectrum? The following is example I found...
Problem Statement : To find the area of the shaded segment filled in red in the circle shown to the right. The region is marked by the points PQRP.
Attempt 1 (without calculus): I mark some relevant lengths inside the circle, shown left. Clearly OS = 9 cm and SP = 12 cm using the Pythagorean...
i can write the equation of circle easy enough, x^2+(y-r)^2=r^2. i get A=r^2/2 * asin((y-r)/r) + (y-r)/2 * sqrt(r^2 - (y-r)^2) through integration (using change of variable). Letting u = (y-r) and u^2=(y-r)^2, du= dy. Here's the rub... it's not right... :-) Appreciate and thanks in...
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Research article (paywall):
https://dx.doi.org/10.1016/j.jpsychires.2022.03.006
Cheers,
Tom
I was asked to derive the relation $$p = u/3$$ for photon gas. Now, i have used classical mechanics and symmetry considerations, but the book has solved it in a interisting way:
I can follow the whole solution given, the only problem is the one about the probability to colide the sphere!. Where...
I just simply used the formula to solve. Note the "x" represents multiplication in this case
0.5 x a x c Sin B
This is based on the conditions given in the textbook I am using which quotes "Use this formula to find the area of any triangle when you know 2 sides and an angle between them"
So I...
I am not very good at proofs. The only thing I have come up with is the following regularity. However, I am not sure how this can be related to the above problem.
Given a sphere ##S_a## with a center ##C## and a diameter of ##a##. I can now construct a line segment ##b## with the endpoints...
Find question here;
Find solution here;
I used the same approach as ms- The key points to me were;
* making use of change of variables...
$$A_{1}=\int_0^\frac{π}{4} {\frac {4\cos 2x}{3-\sin 2x}} dx=-2\int_3^{-2} {\frac {du}{u}}= 2\int_2^3 {\frac {du}{u}}=2\ln 3-2\ln2=\ln 9 - \ln 4=\ln...
I am still looking at this question. One thing that i know is that the
distance ##AB=\dfrac {(λ+1)\sqrt {2λ^2-2λ+1}}{λ^2}##
distance ##OA=\sqrt 2##
distance ##OB=\dfrac{\sqrt{λ^2+1}}{λ^2}##
Perpendicular distance from point ##B## to the line ##OA=\dfrac{\sqrt{2(λ^4+2λ^3+2)}}{2λ^2}##
Therefore...
Below is an image to calculate the surface area of a sphere using dA. I can see how ##rcos\theta d\phi## works, but I don't understand how that side can't just be ##rd\phi## with a slanted circle representing the arc length. The second part I don't understand is why it is integrated from...
I’m wondering if there is a formula for calculating the coordinate points of a polygon given the following
- Center point is known
- area is known
- Point A is known
- Points B, C, and D are UNKNOWN
I am NOT a math pro - this is for a puzzle I’m trying to solve and I can’t remember if this...
I have the solution for this problem using dydx as the area. Worse yet, I cannot find another solution for it. Everyone seems to just magically pick dydx without thinking and naturally this is frustrating as learning the correct choice is 99.9% of the battle...
So, I was curious how one might...
Find the solution here;
Find my approach below;
In my working i have;
##A_{minor sector}##=##\frac {128.1^0}{360^0}×π×5×5=27.947cm^2##
##A_{triangle}##=##\frac {1}{2}####×5×5×sin 128.1^0=9.8366cm^2##
##A_3##=##\frac {90^0}{360^0}####×π×10×10##=##78.53cm^2##
##A_{major...
My interest is on part (c) only.
Wow, this was a nice one! boggled me a little bit anyways; my last steps to solution,
##A= \dfrac{1}{2} ×16.8319^2 × 0.5707=80.84cm^2## bingo!
Any other approach apart from using sine?
Cheers guys