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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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1answer
61 views

Find area of cylinder $x^2 + y^2 = r^2$ that satisfies $0 \le z \le y$

I think I can imagine the shape of surface area. This is what I did: $$\begin{eqnarray*} \text{Surface area} & = & \int_{0}^{\pi} r y \sqrt {2} \, d\theta \\ & = & \int_{0}^{\pi} r\...
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0answers
26 views

How to calculate area of triangle-like structure of blocks? (Pick's Theorem seems insufficient.)

Given the following structure, is there a formula to calculate the number of blocks? (EDIT: and I am really looking for a solution for any BASE and HEIGHT.) At first, it would seem that this is a ...
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0answers
33 views

Finding the area and perimeter [on hold]

Create two rectangles so that the first has exactly twice the perimeter of the second and the second has exactly twice the area of the first. I’ve tried many combinations but nothings worked so far. ...
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1answer
40 views

How do you find area of the loop in the graph of $x(x^2+y^2)=(x^2-y^2)$

The graph of the given equation is $x(x^2+y^2)=(x^2-y^2)$"> I believe I have to use (r,θ) coordinates but I do not know how to integrate this in (r,θ).
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2answers
34 views

Find the area of the triangle $ABC$ in terms of $t$ and $t'$ [on hold]

Problems : Let the triangle $ABC$ be right in $A$ , and the inscribed circle of this triangle touches the hypotenuse $[BC]$ in the point $T$ If we called $BT=t$ , and $CT=t'$ , calculate the area of ...
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1answer
20 views

Differential area for the lateral surface of frustum of a cone

I am studying Fluid Mechanics and I needed a differential area element of the side or lateral surface of a frustum. This frustum is cut from a cone. In solution manual of the book I study, ...
3
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3answers
44 views

Area defined by $x^2+y^2 \leq 1$ and $y\geq x(x^2-16)$

Area defined by $x^2+y^2 \leq 1$ and $y\geq x(x^2-16)$ One very obvious way would be to find the points of intersection which would be messy and subject to many conditions. I was trying to solve ...
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1answer
40 views

Find the diameter of a circle subtended by an angle

The question doesn't state whether its subtended at the center or circumference, but I not sure if it matters The sector a circle subtended by an angle of $22.5$ degrees has an area of $\frac{9\pi}{4}...
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0answers
23 views

Integrating an absolute value function to find area between curves $[∫^{-1.02}_{-2.84}(|cos(5.7x-10)|+1.7)dx] $

I'm trying to find the area between the curves $5.7 e^{-0.7x-3}+1.3 $ and $-|cos(5.7x-10)|+1.7 $ from $-2.84$ to $-1.02$ After graphing this and finding the upper and lower functions, it lead me to ...
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4answers
45 views

Why is my approach for showing $r^2 \frac{\theta}{2}$ equals the area of a circular sector incorrect? Do we need calculus?

I know that the area of the sector of circle can be computed using the following two formulas $$\pi r^2 \frac{\theta }{360} \space \space \text{ (degrees case)}$$ $$or$$ $$r^2 \frac{\theta }{2} \...
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1answer
60 views

Area bound by $y =\ln x +\tan^{-1} x$?

I'm not really sure how to do this without having a real graph using a graphing calculator. Adding the integrals would no work since some areas are overlapping, and even subtracting doesn't since one ...
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1answer
42 views

Finding $\iiint 6z\,dx\,dy\,dz$ over $\lbrace (x,y,z) \in \mathbb{R}^3 : |x+y| \le z \le |x|+|y|\le 1 \rbrace$

I want to calculate integral : $\iiint 6z\,dx\,dy\,dz$. The area is $$\Omega= \lbrace (x,y,z) \in \mathbb{R}^3 : |x+y| \le z \le |x|+|y|\le 1 \rbrace$$ My problem is this, that I don't know what is ...
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1answer
29 views

Determine the area enclosed by the curve

Determine the area enclosed by the curve with two polar equations: $r= \sqrt 2 \sin(\alpha)$ $r^2 = \sin(2\alpha)$ I have no clue how to do this. A formula we are given is the one below but I'm ...
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1answer
40 views

Number of Atoms on Earth's Surface (Question based on Vsauce video which involves fractal dimension)

In a Vsauce video titled "How much of the Earth can you see at once" they try to do a calculation to estimate the number of atoms on Earth's surface. The first part is easy: 1) Calculate surface ...
3
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1answer
52 views

heron's formula proof

I have seen an interesting proof of Heron's formula here. It is very simple, but I do not understand one point. The author demands, that the formula should contain factor $(a+b+c)$, because when we ...
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0answers
20 views

Determining area of graph based on unknown boundaries bounded by special trigonometry graphs

Interesting question for all to solve that i chanced upon luckily. let the area bounded by two graphs below from $m \pi $ to $(n-1) \pi $ be A, where $m$ and $n$ are integers. Express $n-m$ in terms ...
2
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0answers
20 views

Find sub areas of a function in a circle

I have a cellular signal calculation function, which calculates the signal given the distance from the antenna. Without the constants, the function is basically: $f(d)=1/(d^α)$ where α is a parameter. ...
1
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1answer
67 views

Evaluating Area Under a Curve [closed]

Consider the curves: $$\displaystyle f: \Big(\frac{x-c}{a}\Big)^{2k+1} + \Big(\frac{y-d}{b}\Big)^{2k+1}=1$$ $$\displaystyle g: y = -\frac{b}{a}x +d+ \frac{bc}{a}$$ Where $a$,$b$,$c$,$d$,$k$ $...
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1answer
47 views

Area inside $r = 1 + \cos \alpha $

We are studying polar equations. Calculate the surface inside $r = 1 + \cos \alpha $ and outside $r = 1$. I know the area inside $r = 1 + \cos \alpha $ being $\frac{3 \pi}2$ because I calculated $\...
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3answers
81 views

What is the area of a triangle with sides $\sqrt{5}$, $\sqrt{10}$, $\sqrt{13}$?

I found a "fun algebra problem" that asks you to find the area of a triangle whose sides are $\sqrt{5}$, $\sqrt{10}$, $\sqrt{13}$. After some algebra hell trying to work with Heron's formula, I ...
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2answers
46 views

A given perimeter length that is circular encloses the maximum area - which are the (analytic) proofs? [duplicate]

I'm guessing Newton, because of his integrals. But what proofs have been established, and which is the most mathematically intuitive one? I was looking for the tag "circumference", supplied the newer ...
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4answers
77 views

Finding the area between two curves.

Context: High School question. Find the surface area between the curve of the function $y=6-3x^{2}$ and the function $y=3x$ in the interval $[0,2]$ My approach: -We must find the points of ...
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0answers
14 views

Proving an integration identity using the co-area formula

I want to prove that identity: $$\int _{B^n(0,R)} f(x)dx = \int _0^R r^{n-1} \int _{S^{n-1}} f(ry)dS(y)dr$$ where $B^n(0,R)$ is $n$-dimensional ball and $S^{n-1}$ is the $(n-1)$-dimensional sphere (...
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1answer
31 views

Maximizing the size of n similar equilateral triangles from a rectangle

I have 3 rectangles of greenhouse sheeting material. They are each 12 feet long and 4 feet high. I want to use this material to clad a 3/4 icosahedron dome structure. What that means is that I will ...
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3answers
62 views

Solve $\int^2_{-1} (1-x)dx$ by thinking in terms of area?

On our last quiz in Calculus 1, my professor asked us to solve $\int^2_{-1} (1-x)dx$ by thinking in terms of area. I don't know what he means by, "thinking in terms of area". I can solve it myself, ...
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0answers
28 views

Polar area and do I need absolute value signs?

Imagine a curve such as $$r=\sqrt{\sin x}$$ The area over $(0,\pi)$ would be calculated as follows: $$\int_0^{\pi}\frac{1}{2}\sqrt{\sin x}^2dx=\int_0^{\pi}\frac{1}{2}|{\sin x}|dx, not \int_0^{\pi}\...
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0answers
25 views

Area between a parabola and a line

Hello, can someone explain me, or at least give me some instructions, in order to understand why is the surface of the arc given by that integral? I've done some research and nothing that i do is ...
3
votes
1answer
36 views

maximum area of inscribed quadrilateral

Find the maximum area of the quadrilateral inscribed on $y=2x-x^2$, where $y\geq 0$ and explain your answer. I can just estimate the shape but I don't know how to prove it precisely. Help me with a ...
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1answer
28 views

Proving there exists a hypercube inscribed in an area limit by a curve

I am facing a problem as follows. Given a set $A = \{X \mid X \in R^n, 0 \le f(X) \le \xi\}$, where $X=[x_1,x_2,\dots,x_n]^\top, \xi > 0$, $f:R^n \rightarrow R$ is a continuous function. Can we ...
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2answers
88 views

Maximize area of a quadrilateral given three sides

What is the maximum possible area that a quadrilateral can have, if the lengths of three of its sides are given as 3, 4 and 5, while the fourth side can have arbitrary length? (Thinking of it as three ...
2
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0answers
32 views

Calculate the area bounded by $x^2+y^2=(\frac{x}{a})^3+(\frac{y}{b})^3$ and $x=0,y=0$

Calculate the area bounded by $x^2+y^2=(\frac{x}{a})^3+(\frac{y}{b})^3$ and $x=0,y=0$ We may assume $a,b>0$. Use the polar coordinate it's equivalent to value $\int_{0}^{\frac{\pi}{2}}\int_0^{\...
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1answer
27 views

Find area using double integral and polar coordinates

Find the area enclosed by $ρ=1+cos(\theta)$. I can not find the angle of the function to define the limits of the integrals. This would be the graph of the function: What I was trying to do, because ...
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1answer
17 views

How can I find the volume generated by revolving the following region about $x=5$?

The region enclosed by : $y=6-x^2$ and $y=5$ I first get the inverse functions and the intersections and then work with the disk/washer method, the result is zero and I can't figure out what am I ...
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3answers
29 views

Calculate the area of the curve $\cfrac{e^x}{e^{2x}+9}$ between the x-axis

6.Calculate the area located on x-axis and below the curve $y=\cfrac{e^x}{e^{2x}+9}$ I've thinking of finding the intersection points of the curve and $y=0$ \begin{align} e^x& = 0 \qquad /\ln ...
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1answer
14 views

Triangle with two sides given find the greatest area if the area is prime number [closed]

A triangle has two sides of lengths 4 centimeters and 6 centimeters. Its area is n square centimeters, where n is a prime number. What is the greatest possible value of n? (A) 11 (B) 12 (C) 19 (D) 23 (...
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1answer
52 views

Find the region bounded by $y=x \sin x$, and $y=x$

Find the area bounded by the region $y=x \sin(x)$, and $y=x$, for $0\le x\le \frac{\pi}{2}$. My attempt Area $=\int_\limits{0}^{\frac{\pi}{2}}(x-x\sin(x))dx$ After integrating I got: $$[\frac{x^2}{...
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1answer
20 views

Given a set of points, find maximum area of triangle

Given a finite set of 2-d points, I need to find the maximum area of triangle formed. My solution steps : Take mean of points , lets call it (x_m, y_m). Take 3 most distant points from (x_m, y_m). ...
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2answers
63 views

Area below the curve $y=\left[\sqrt{2+2\cos2x}\right]$

Find the area below the curve $y=[\sqrt{2+2\cos2x}]$ and above the $x$-axis , $x\in [-3\pi,6\pi]$, (where $[.]$ denotes the greatest integer function) . My approach: $$y=[\sqrt{2+2\cos2x}]$$ $$=[\...
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1answer
104 views

Finding the area enclosed by curve defined by $\arcsin x+\arcsin y=\arcsin(x\sqrt{1-y^2}+y\sqrt{1-x^2})$

If $\arcsin x+\arcsin y=\arcsin(x\sqrt{1-y^2}+y\sqrt{1-x^2})$ Then the area represented by the locus of point $(x,y)$ if it is given that $|x|,|y|\leq 1$ My Try: Put $x=\sin \alpha$ and $y=...
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0answers
36 views

Uncoditional formula for unsigned area under a linesegment w.r.t. $y=0$

I'm interested in finding the unsigned area under a line segment with respect to $y=0$. The line segment is defined by start point $(s_x, s_y)$ and end point $(e_x, e_y)$ Without loss of generality, ...
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1answer
36 views

I have no idea how to solve this problem using areas of known cross section

The problem involving cross sections I am so confused on how to find volume using known cross sections. I've never understood it. This problem that I've encountered is very difficult, and I tried ...
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0answers
11 views

Find a minimum (polygonal) bounding box for a closed curve

This is a hard question to ask, I think. Please just try to put me on the right track, at the very least. Let's assume that I have a closed curve, with x number of edges. I'm able to find the ...
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1answer
24 views

Proof of area function equals antiderivative anomaly

In my math textbook they have a proof that an antiderivative to f(x) is the same as the area function to f(x). They assume that f(x) is continuous and that f(x) is a growing function. By looking at ...
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2answers
194 views

O is a point in triangle ABC. OA, OB and OC are joined and produced to meet BC, AC and AB at D, E and F. Find the value of OD/AD+OE/BE+OF/CF.

O is a point in a triangle ABC. OA, OB and OC are joined and produced to meet BC, AC and AB at D, E and F. Find the value of OD/AD+OE/BE+OF/CF. I took the special case when O is exactly in the centre ...
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1answer
44 views

Surface area of a sphere dilemma

I recently found that surface area of a sphere can't be found with the following method. What's the flaw in it? First, I have taken a very thin ring of thickness $dx$ at a distance of $x$ from the ...
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4answers
33 views

Calculating a circle area by rotating its diameter

I thought a circle as a set of dots . The circumference is about 314.16 dots long and the diameter 100 dots long. I am wondering if it is possible to calculate the area of a circle by rotating its ...
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vote
2answers
58 views

Area of a rectangle inside a triangle with given coordinates

Given a triangle with vertices at points $(0, -a), (0, a), (b, 0)$, where $a > 0$, find the maximal area and the dimensions (base and height) of a rectangle that can be contained within the ...
4
votes
5answers
170 views

Finding Angle using Geometry

In an equilateral triangle $ ABC $ the point $ D $ and $ E $ are on sides $ AC $ and $AB$ respectively, such that $ BD $ and $ CE $ intersect at $P$ , and the area of the quadrilateral $ ADPE $ is ...
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0answers
20 views

Area of Convex polyhedron (2D) with unordered vertices

I am aware that an algorithm exist to find area of a convex polyhedron when the vertices are given in order. But, I have a convex polyhedra which does not have vertices in order and I wish to compute ...
0
votes
1answer
35 views

Area of Parallelogram in $\mathbb R^n$

Let $\{u,v\}\subset\mathbb R^n$ be linearly independent. Then $u$ and $v$ induce a parallelogram. If $n=2$, then its area is $|u_1v_2-u_2v_1|$. If $n=3$, then its area is $\|u\times v\|$. Is there ...