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Questions tagged [area]

Area is a quantity that expresses the extent of a two-dimensional or three-dimensional surface or shape.

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62 views

Area of Rectangle and circle

Just a basic question : The area of a rectangle is length×width because I'm filling the length with width. This makes sense to me but if I do the same approach for a circle then the area becomes $2\...
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2answers
81 views

What is the area formed by the loop of the curve $x^3 - y^3= xy$?

I know how to find area of elementary functions by integration. The only loop questions that I know how to solve are the ones that are of the petal type, which can be converted into polar form and ...
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1answer
61 views

Area of sector bounded by line and curve

I have a line segment $\overline{AB}$, with endpoints $(x_1,y_1)$ and $(x_2,y_2)$. Drawing any function $f(x)$through those 2 points, the function must have an average velocity over $[x_1,x_2]$ of $\...
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1answer
49 views

Determine the area bounded by the curves $y=2x$ and $y=x^2$ and the parabaloid $z=x^2+y^2$

Determine the area bounded by the curves $y=2x$ and $y=x^2$ and the parabaloid $z=x^2+y^2$ I'm not really sure what my function that I'm integrating is supposed to be I believe I want to integrate on ...
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1answer
59 views

How can I determine the area of the region bounded by the curve $x^2=y^4(1-y^3)$ [closed]

$x^2=y^4(1-y^3)$ On a graph it looks like a diamond. This is for my integral calculus class.
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0answers
87 views

Bounded Area between $y=x^2$, and $y=9$, and $y=k$

I have been given a function to find the horizontal line $y=k$ which would splice the area bounded between $y=x^2$, and $y=9$, into two equal parts. I approached the function using symmetry to find ...
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1answer
16 views

How to tell the run of a line segment where the area underneath it, starting point, and slope are known.

I have a program where I am currently solving for the difference of X₁ and X₂ of a line-segment where the area under the line-segment (A), the slope (s), and the starting point (X₁,Y₁) are known. ...
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1answer
14 views

Surface area of ellipsoid created by rotation of parametric curve

I have a parametric curve (elipse) defined as follows $$\begin{aligned} x(t) &= \cos(t)\\ y(t) &= 2 \sin(t)\end{aligned}$$ and need to calculate the surface area of the ellipsoid produced by ...
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1answer
49 views

Solid of revolution using washer method(Gives negative answer)

My teacher even knew that the answer should not be negative but it turned out to be negative. The given was y=x^2, y=4x-x^2, revolving about the y-axis. Here are some of the solution presented, I hope ...
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1answer
56 views

Area of the surface

Compute the area between the surfaces $(x^2+y^2)^2=a^2(x^2-y^2)$ and $x^2+y^2+z^2=a^2.$ My attempt: I take the first case $z\geq 0$: $$x=x$$ $$y=y$$ $$z=\sqrt{a^2-(x^2+y^2)}$$ then $$T_x=(1,0,\...
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1answer
52 views

Finding enclosed area

Hints only please !! I gave a test today, and it was asked to find area enclosed by the curve $x^4 + y^4 = 2*x*y$. This is an implicit function. A quick obs. shows me that, the curve is entirely bound ...
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3answers
26 views

Under between the x- and y-axis of the graph of $y=x^n$

Consider the graph of $y=x^n$ for $n>1$ and $x>1$. The area bound between the curve and the x-axis between $1$ and $a$ is one third the area between the curve and the y-axis between the values ...
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0answers
37 views

How to formally prove this integral identity?

I was doing some investiations into 'areas under curves' an found by a geometric analogy the following integral identity: $$ \int_0^a f(x)dx =\frac12 \int_0^{f ′ (a)} (a- f^{-1}(f′(x)))^2 dx $$ ...
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2answers
68 views

Finding out the value of $\angle DQC$ in a trapezium $ABQD$ where $\angle DCB$ = 30$^\circ$

In this below diagram, $\angle ABC=60^\circ, \angle DCB=30^\circ $, $AD$ is parallel to $BC$ and $AP$ is perpendicular to $BC$. Both the area and perimeter of $ABCD$ and $APQD$ are equal . What is the ...
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1answer
42 views

Improper integral - Calculating Area of Floor Function

I'm stuck trying to calculate the area of the region defined by: $\begin{Bmatrix} {(x,y)\in\mathbb{R^2}:0\leq x\wedge 0\leq y\leq 2^{-\lfloor x \rfloor}} \end{Bmatrix}$ I'm just starting studying ...
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3answers
37 views

Area of a right trapezoid with bases $15$ and $8$, and height $9$

I can't figure out how they got that. I got 1080, they got 103.5. How???
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1answer
22 views

Please explain how to take limits in double integral while finding volume using the given problem

Question: Find the volume under the surface $z=\sqrt{1-x^2}$ and above the triangle formed by $y=x$, $x=1$ and $x$ axis. The two integrals are given as follows: $$\int_0^1 \int_y^1 \sqrt {1-x^2} \,...
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0answers
30 views

Generalizing computation of number of pixels in inscribed circle to axis-aligned ellipses at arbitrary points

This answer really nicely sums up the question of how to compute the number of pixels inside an inscribed circle. However, I am looking for a more generalized version of this in two ways. 1) I would ...
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2answers
73 views

Proof for area of an equilateral triangle with respect to one side?

I'm trying to find out the area of an equilateral triangle with respect to one side. Anything wrong with my proof? An equilateral triangle with sides of length $a$ can be divided in half along the ...
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1answer
46 views

Inequalities about area and perimeter

"A gardener is laying out a rectangular lawn. His specifications are that the area $(A)$ must be greater than $40$cm but the perimeter $(P)$ must be less than $40$cm. if the width of the lawn $(w)$ ...
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1answer
32 views

Help indentifying unknown method of computing area by rolling a ball about the perimeter

Some time ago I saw a brief presentation about a newly discovered method used to compute the area of certain two-dimensional shapes by rolling a circle (of varying radius) about the perimeter; the ...
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1answer
25 views

Is this the simplest way to visually prove the “scalene trapezoid area formula”

Please refer to image. Is there a simpler way to visually prove the scalene trapezoid area formula?
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1answer
37 views

Find area of the fourth triangle given the area of three triangles.

This is the question that I got in TCS Ninja under the Quantitative section. How shall I do this? Help !!
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0answers
122 views

What is the area of this figure?

I want to know how to find the area of this shape: Yellow, white and blue shapes are ellipses. Red is a square. The blue ellipse is not cut in half by the square. I know that I have to add up all ...
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1answer
55 views

Area of a circle segment on sphere, given radius (meters) and central angle (degrees)

Situation I have a circle segment and some information about the circle it belongs to. Given Information: radius of the circle in meters central angle in degrees lat/long of all three points on the ...
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1answer
64 views

How to double the circle?

I'm looking for a compass-and-straightedge method to construct a circle that has area twice of the area of another circle, with no prior knowledge of π, without knowledge of the formula for the area ...
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0answers
50 views

How do I find the area bounded by a $y=\frac{4}{3}x^2+\frac{12}{3}x-3$ and $y=\sqrt{x}$

I am having difficulties with this problem:$y=\frac{4}{3}x^2+\frac{12}{3}x-3$ and $y=\sqrt{x}$. Graphng two of the functions I get the following: Graphing both functions shows that they intersect at $...
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0answers
26 views

how to calculate wall thickness of a mesh?

based on similar questions on mesh volume, volume of a mesh can be calculated by following equation: volume = ((vec1 x vec2) . vec3) /6 where vec1, vec2, and vec3 are the vectors from origin to a ...
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1answer
106 views

Determining the area of a right triangle, perimeter given, hypotenuse value given in terms of one of the legs.

The problem states: Right Triangle- perimeter of $84$, and the hypotenuse is $2$ greater than the other leg. Find the area of this triangle. I have tried different methods of solving this ...
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1answer
33 views

Find region of $0\le y\le1$ and $y\le x\le1$ [closed]

How i can find the region that is bouned from these inequalities? Any general rule when we have to deal with these inequalities? Ploting the graphs?
2
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2answers
57 views

Function whose graph is dense in the plane [duplicate]

Is there a function $f:\mathbb R\to\mathbb R$ such that for every disc in $\mathbb R^2$ the graph of that function has at least one point that lies inside that disc? I searched for something similar ...
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1answer
122 views

Help find work done to pump fluid

A tank is full of oil weighing 20 lb/ft${^3}$. The tank is an inverted right rectangular pyramid (with the base at the top) with a width of 2 feet, a depth of 2 feet, and a height of 5 feet. Find the ...
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2answers
181 views

Orthogonal projection of an ellipsoid [closed]

Suppose an ellipsoid given by $\{{(x,y,z)| {x^{2}+\frac{y^{2}}{4}+\frac{z^{2}}{9}}}=1\}$, find the area of the orthogonal projection of the ellipsoid on the plane ${2x+4y-5z=10}$. What is the right ...
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3answers
120 views

How to determine the area of a rotated ellipse?

The ellipse $6x^2+4xy+5y^2+8x+8y+1=0$ is neither expressed in terms of $x$; like $y=\pm\sqrt{a^2-x^2}$, nor in terms of $y$; like $x=\pm\sqrt{a^2-y^2}$. Separation of $x$ (or $y%$) may be impossible. ...
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0answers
73 views

How to calculate the area and volume of a random 3d shape knowing only its coordinates?

Given a set of 3d points which make up an vector object of any shape (with any number of points), without the edges being known, how can the object's edges be found/detected so that the object's ...
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2answers
35 views

Area that is bounded by functions

The functions $$f_k(x)=\frac{x+k}{e^x}$$ are given. Let $A(u)$ be the area that is bounded by $f_1, f_3$, the $x$-axis und the line $x=u$. I want to check the area if $u\rightarrow \infty$. $$$$ ...
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1answer
44 views

Calculate Overlapping Area of $2$-Dimensional Shapes

I am running a Computer Simulation where 2 Shapes are moving towards each other and will eventually overlap. I want to calculate the overlapping Area of the shapes - in this example a Circle and a ...
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2answers
91 views

Find area of quadrilateral in triangle. [closed]

What is the area of $HIJK$ quadrilateral, if the area of $ABC$ triangle is $70$, $BE=ED=DA$, and $BF=FG= GC$?
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3answers
56 views

Finding percentage increase of area [closed]

Find the percentage increase in the area of a triangle if its each side is doubled?since no information is given about the type of triangle in question, so should i take a equilateral or isosceles or ...
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2answers
96 views

Do the given perimeter and area corresponds to many shapes? [closed]

I have a perimeter P and area A of a planar shape. How to prove that there are many shapes that corresponds to those perimeter and area values?
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2answers
54 views

How do you calculate area of two circles joined by tangent lines

please can you help to provide the mathematical steps required to calculate the area of a shape formed by two circles of different diameter joined together by two tangential lines.
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1answer
70 views

Area of the region

Let $O(0,0), A(3,0), B(3,2)$ and $C(0,2)$ be four vertices of a rectangle. Let $$d(P,OA)≤\min {\Bigl(d(P,AB),d(P,BC),d(P,OC)\Bigl)}$$ where $d$ denotes the distance of the point $P$ with the line ...
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0answers
11 views

Area in d dimensions of a spherical part [duplicate]

Let $S^{d-1}$ be the unit sphere (centered at $O$) in $d$ dimensions. One can show that when $d=3$, for fixed $x\in S^{2}$ the area of $P(x) = \{y\in S^{2}, \angle xOy < \alpha \}$ is $2\pi(1-\cos\...
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1answer
23 views

Regarding Areas of volumes

I have a question: In general, area of square is Length times Length. If the length is less than $1$ then do we end up with less area? Like if we have a side whose length is $0.5\mathrm{m}$. Then ...
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2answers
54 views

BdMO - 2018 Regional - Geometry 9 [closed]

In $ABCD$ tetragon $E $ and $F $ are mid points of $AB$ and $AD$ respectively. $CF$ intersects $BD$ at point $G$. If $\angle FGD = \angle AEF$ and the area of $ABCD$ is $24$ , what is the area of $...
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1answer
84 views

1000cm^3 of metal is to be cast as a rectangular block with square ends (Calculus Optimization question)

1000cm^3 of metal is to be cast as a rectangular block with square ends. Use calculus to show that for the least surface area a rectangular the rectangular block needs to be a cube. Having some ...
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1answer
43 views

Area of a triangle in space using determinants from 3 points

I know there is a way to find the area in plane using 3 points but when it comes to space(3D) does it work too? if so how should it look? Thanks
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1answer
156 views

Calculate surface normal and area for a non-planar quadrilateral

Given the four coordinates of the vertices, what is the best possible approximation to calculate surface area and outward normal for a quad? I currently join the midpoints of the sides, thus ...
2
votes
1answer
49 views

Calculating integral over unit ball using co area formula

Calculate $\int_V\frac{1}{2+(x^2+y^2+z^2)^{\frac{3}{2}}}dxdydz$, where $V=\{(x,y,z)|x^2+y^2+z^2\lt 1\}$ using the co area formula. So I know that the formula is: $\int_V f dx$ = $\int_a^b\int_{M_c}\...
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1answer
46 views

find the area of the octagon [closed]

Find the area of the colored octagon (tell the ratio between square $ABCD$ and the colored polygon). Square $ABCD$ is a perfect square, and $E,F,G,H$ are the midpoints of the line they are ...