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
41 views

Maximum ratio of the area of two regions?

A regular hexagon is split into two regions by a straight line so that the ratio of the perimeters of these regions is 2:1. Find the maximum ratio of the areas of the two regions. This problem is ...
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Minimal surfaces

Among the definitions of minimal suraface I found these two: (1) A surface $M\subset\mathbb{R}^3$ is minimal if for any point $p\in M$ there is a neighborhood $U$ of $p$ in $M$ that minimizes the ...
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Finding the Area bounded by 1 function and 2 lines

Find the area bounded by $y = e^{x}$, $y = e^{2}$ and $x = 0$ I found the area by integrating the function $e^2 - e^x$ with the x-bounds of $0$ to $2$, which turned out to be $e^{2} + 1$. $$\int_0^...
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4answers
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How to show that $\int_{0}^{1}\sqrt{x}\sqrt{1-x}\,\mathrm dx =\frac{\pi}{8}$

I was reading Advanced Integration Techniques, and found that$$\int_{0}^{1}\sqrt{x}\sqrt{1-x}\,\mathrm dx =\frac{\pi}{8}$$ The book provides one method using residue theorem and Laurent expansion. ...
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1answer
52 views

Area under the graph of a function defines Integration or Integration defines area under the graph of a function?

What will be the area under the curve x^(-1/3) in the interval [-1, 1]? Is this area finite? Does definite integration in any interval always computes area under the curve of the function in that ...
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2answers
41 views

Let $a,b,c$ be the sides of a triangle. The find maximum value $ \frac{A}{S^{2}}$

Let $a,b,c$ be the sides of a triangle, $A$ is the area and $S$ is the semi-perimeter $(a+b+c)/2$. Find the maximum value $\frac{A}{S^{2}}$. My Approach: Method 1: Applying AM-GM inequality on $...
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2answers
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Three cevians in a triangle create four sub-triangles of area $1$. Find the area of a non-triangular region.

I'm having trouble proving that all the white and green areas have the same area, from there on we can obtain the answer $1+\sqrt5$ by proving that the inner red triangle points are midpoints.
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Finding the ratio of 2 sides, and expressing OP in terms of a, b, c

I got a bit confused about this problem, and its something that my country's academic curriculum does not really delve into or practice as much. The Problem: Let $A,B,C$ be three points on a plane ...
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Integral over inersection of sheared and dilated cirlce with another circle.

I am currently working on a project relating to the topic of Wavelets and Shearlets. I am wanting to estimate a Shearlet transform of the indicator function of the ball of radius $r$ in $\mathbb{R}^2$....
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28 views

A machinist is required to manufacture a circular metal disk with area $2000 cm^2$.

A machinist is required to manufacture a circular metal disk with area 2000 cm^2. (a) What radius produces such a disk? (Round your answer to four decimal places.) Answer; 25.23132522 <----- This ...
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Show if area of surface is finite or not

Let $S=\{(x,y,z) \in \mathbb R^3 \mid xyz=1\, x,y \in (0,1)\}$.The question is, is the area of this surface finite or not? First of all I choose a parametrization of S. Let $\sigma:(0,1) \times (0,1) \...
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Integration of multi-valued function

Graph I have a set of discrete (x,y) points and need to find the area between the curve and the y-axis. However, the curve seems to be a multi-valued function (multiple x for the same y). Is such an ...
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Find area between $f(x)=\begin{cases}x^2-1,&x<1,\\\ln(x),&x\geq1\end{cases}$ and $x$-axis to the right of the absolute extrema

Given $$f(x)=\begin{cases}x^2-1,&x<1,\\\ln(x),&x\geq1,\end{cases}$$ find the area between $f(x)$ and $x$-axis to the right of the absolute extremum. The $x$-axis is the equation $y=0$. We ...
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1answer
35 views

Find the ratio of the area $EDPQ$ to the area of ABC

In triangle $ABC$,points $E$ and $D$ are on side $AC$ and point $F$ is on side $BC$ such that AE=ED=DC and $BF:FC$ =2:3. $AF$ intersects $BD$ and $BE$ at points $P$ and $Q$, respectively. Find the ...
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815 views

Finding the slope of a line that cuts an area in half

A region $A$ in the first quadrant is bounded by $y=x^2$, $y=25$ and the $y$-axis. Find the value of $m$ with the property that the line $y = mx$ divides $A$ into two regions with the same area.
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Find the area between functions $x^3=2y^2;x=0;y=-2$

Take the element of area parallel to the y axis: $$x^3=2y^2;x=0;y=-2$$ First, I isolated in terms of $y$, $$y= \pm \sqrt{x^3\over2} \\ = \pm{\sqrt2\over 2}x^{3 \over 2}$$ Since bounded by $y=-2$, ...
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1answer
481 views

Road Shape for Square Wheels

Let's say you have a bike with square wheels of side a. In order for a smooth ride, there must be these bumps in the road. Is there a formula for the area of each bump using a?
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61 views

Calculating the area between two functions expressed in polar coördinates

I have the following to polar coördinates: $r=1+\cos(\theta)$ and $r=3\cos(\theta)$. The question is to calculate the area in side $r=1+\cos(\theta)$ and outside $r=3\cos(\theta)$. I know I need to ...
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23 views

Area of a region given in polar coordinates

Determine the area of the following regions given in polar coordinates: $A)$ $D=\{(r, \theta):\, 1+\cos \theta \leq r \leq 3\cos \theta \}$ $B)$ $D=\{(r, \theta):\, r\leq 3\sin \theta,\, r\leq -5.2\...
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Area between $t^2$ and $\sqrt{t}$ in interval [0,2]

This was a question of a exam that I take today. I would like to know the area between $t^2$ and $\sqrt{t}$ when t ∈ [0,2]. My answer was $\frac{1}{6}$, since there is no common area when t>1.
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Area included in the curves [closed]

What is the area included in the curves √x + √|y| = 1 and |x| + |y| = 1. I know the area the total area of the |x| + |y| = 1 is 2 units square but i cannot determine the area of the given first ...
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48 views

Find the value of $a$,the value of $b$ and the value of $c$

A rectangle has sides of length $x-3$ units and $ax^2+bx+c$ units,where $a,b,c \in \mathbb Z$. The area of the rectangle is $2x^3-13x^2+25x-12$ square units. Find the value of $a,$ the value of $b$ ...
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Calculating the force acting between to magnets [migrated]

I am using neodymium N52 magnets in a project I'm working on, and I have been attempting to calculate the force between them. I just wanted to double check my results with the community and hopefully ...
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1answer
82 views

How to calculate an area under $y=x^{-2}$ without integral

I need to get a formula for area under $y=x^{-2}$ for $x \in (1,a)$, where $a \in (1, +\infty)$, WITHOUT using integrals. I tried following: Let $h=\frac{a}{n}$, where $n$ is natural number of ...
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31 views

Getting the AREA of an OIL SPILL

How much area in $m^2$ will $200 cm^3$ of oil cover if it forms a layer of $35mm$ thick? i dont know what formula i will be using, my teacher never discussed this and i cant seem to find any on the ...
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3answers
59 views

Area between $x^3$ and $\sqrt[3]{x}$

I'm having some difficult to find the intersection between $x^3$ and $\sqrt[3]{x}$ for calculate the area between them. Could someone help me?
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1answer
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Primary school competition problem find the area of a square

Someone posted online a primary school competition problem. I solved it using the Cayley-Menger determinant (finding the answer is 169/2 after getting the length of AD being 13) but it probably isn't ...
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1answer
20 views

Area moments of a Freeman chain contour

The area moments of a polygon can be computed by generalizations of the shoelace formula for area. In particular, the first and second order moments are given by https://en.wikipedia.org/wiki/...
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787 views

Area of Circle Overlapped by Rectangle

I'm trying to determine 'how much' (as a percentage) a 2D rectangle fills a 2D circle. Actual Application: I was comparing the accuracy of some computer game weapons by calculating the max possible ...
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952 views

Area of a circle $\pi r^2$

So, today I learned that the area of a circle is $\pi r^2$. So, I thought that since $r$ is $1$ dimensional, $r^2$ will be $2$ dimensional. In this case, a square, as you only multiply $2$ dimensions (...
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Area - Square of Perimeter Ratio

I am currently working at a GIS (Geographical Information Systems)-related project where we have a bunch of pieces of land. I want to study whether the shape of each piece influences a target variable....
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1answer
42 views

Find the minimum value of the perimeter

Find the minimum value of the perimeter of a triangle whose area is 3 $cm^2$ I tried it using Hero rule $$A = \sqrt{s(s-a)(s-b)(s-c)}$$ $$9 = s(s-a)(s-b)(s-c)$$ But it did not serve me well ?
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1answer
59 views

How to find an area of a figure that's restricted by $\frac{a^3}{a^2+x^2}, a\neq0, a\in R$ and $2ay=x^2$ using definite integral

I have to find an area using definite integral, where the figure is restricted by two functions, stated in title. Can someone explain how to do that? Here is my try: So I equal the expressions to ...
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In $\triangle ABC$, we have $\angle BAC = 60^\circ$ and $\angle ABC = 45^\circ$… [closed]

In $\triangle ABC$, we have $\angle BAC = 60^\circ$ and $\angle ABC = 45^\circ$. The bisector of $\angle A$ intersects $\overline{BC}$ at point $T$, and $AT = 24$. What is the area of $\triangle ABC$? ...
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1answer
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Ratio of area of a triangle to that of its medians

Let a triangle ABC have medians namely l,m,n. I can't figure out how to find the ratio of area of triangle LMN to area of triangle ABC. LMN is the triangle with sides of lengths of the three different ...
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Find the area of triangle $SVW$

To find length of side $SV$ I used Pythagoras theorem which gave $SV=17$. To find angle $SVW$ I added $45$ to $90 = 135$ to find side length $SW = 17^2+24^2-2 \cdot 17 \cdot 24 \cdot \cos135=37$ and ...
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Ratio between the width of the intersection of two identical intersecting circles and radius, when the intersection is $\frac{\pi r^2}{2}$

Or more visually, if all sections of the below diagram were equal in area and the circles are identical, what is the ratio of s and r, or what is s in terms of r. I came up with an equation using ...
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4answers
121 views

Given triangle $ABC$ with $CE=6 cm $ , $BD=9 cm $ , $\angle A = 60$ , CE , BD are medians. Find the area of $ ABC $

Given triangle $ABC$ with $CE=6 cm $ , $BD=9 cm $ , $\angle A = 60$ , CE , BD are medians. Find the area of $ ABC $ This problem is too difficult for me , the teacher said it is a challenge ...
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1answer
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How is land area calculated when the ellipsoidal shape of the Earth cannot be neglected?

I was curious as to how the land area of a state such as Colorado could be calculated. I understand the area of a 2D rectangle can be calculated using the formula width times length. However, I was ...
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Parametric integration negative area?

I know there is a question very similar to mine already here Why does using an integral to calculate an area sometimes return a negative value when using a parametric equation? , but I am still a bit ...
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Area of Portion of Sphere from a inside Cube

The late painter Maqbool Fida Husain once coloured the surface of a huge hollow steel sphere, of radius $1$ metre, using just two colours, Red and Blue. As was his style however, both the red and blue ...
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Is there a minimal graph in $\mathbb{R}^3$ which is not area-minimizing?

Let $\Omega\subset\mathbb{R}^2$ be an open subset such that $\partial\Omega$ is a closed, simple curve. I'm trying to find an example of an $u:\overline{\Omega}\to\mathbb{R}$ such that $\Sigma:=\...
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110 views

What would be the area of this Red Marked points? And how to calculate this?

I have been given the length $L$ and the width $W$ of a rectangle and the radius $R$ of circle which is situated in the center of the rectangle . I need to find the area of the red marked portion. ...
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What does $\int_\mathbb{R^d}$ means in terms of limits

This might be a silly question, but please help me understand that does the limits of the following expression. The left-hand-side is the actual expression, while the right-hand-side is my ...
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1answer
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Find area of cross section of cylinder by the plane $x$

I am working on my scholarship exam practice (assume high school/pre-university math background) and I think I got half way through but I am not sure how I could continue. Let $r$ be a positive ...
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2answers
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Triangles area related problem

The question is :- In $\Delta ABC$ , $X$ and $Y$ are points on the sides $AC$ and $BC$ respectively .If $Z$ is on the segment $XY$ such that $ \frac {AX}{XC}=\frac {CY}{YB}=\frac {XZ}{ZY}$ ....
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1answer
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Maximum total area of n non-intersect circles?

Given n points on the x-axis, we give arbitrary radius for each point such that each constructed circle doesn't overlap another constructed circle from another point. Which means these circles do not ...
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Area of $y^2=x^2(a^2-x^2)$

Im trying to find the area of $y^2=x^2(a^2-x^2)$ where $a>0$. From my calculations it seems to be $$ 2\int_{0}^{a} x\sqrt{a^2 - x^2}dx=\frac{2}{3}a^3, $$ but I am not sure if it is right.
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Area of a Polygon in a Polygon

I'm dealing with a regular polygon with 7 corners. In this polygon is another polygon defined by connecting one point with the two opposite points of the same polygon. I made a small sketch of the ...
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2answers
472 views

Area of the region $\{(x,y):0\leq x \leq 1, 0 \leq y\leq 1, 3/4\leq x+y\leq 3/2\}$

Find the area of the region $\{(x,y):0\leq x \leq 1, 0 \leq y\leq 1, 3/4\leq x+y\leq 3/2\}$ (using definite integration). I cannot understand how to find this area. I have graphed the lines and found ...