# 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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### An amazing property of the Catenary

I discovered that if we want an arc of catenary in the interval $[a,b]$ we solve $$\int_a^b \sqrt{\cosh '(x)^2+1} \, dx=\int_a^b \cosh x \, dx$$ which means that the "result" of the length is equal ...
1answer
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### Largest rectangle bounded under a function

Let $f$ be a positive monotonically increasing real function in $[0,1]$. Let $F$ be the area under the curve of $f$ ($F=\int_0^1{f(x)dx}$) For every $x\in[0,1]$, let $G(x)=f(x)\cdot (1-x)$ = the area ...
1answer
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### Bisection of a a triangular area

Here's the sketch: From the inner point P of a triangle ABC the three connecting lines to the corner points are drawn. In addition, the lines PE, PD and PF are each drawn parallel to a median of ABC. ...
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### The area enclosed by curve $r^2=9\cos 5\theta$

Find the area enclosed ny the curve $r^2=9\cos 5\theta$. I have drawn its graph and it is the rose which has 5 petals. In order to find the total area it's sufficient to find the area of one petal. ...
2answers
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### Help needed in understanding Heron's Formula

So i just started learning Trigonometry seriously and something doesn't feel right to me, either I'm missing something or not but. Lets assume we have a triangle and there are two ways to find the ...
1answer
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### Tom Apostol - Calculus Vol. 1: Method of Exhaustion in Introduction

I'm new to proof based math, and I really want to get better from pursuing the discipline by myself. I actually had a fear, last year, of proof-based math after my first semester in college. It was ...
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### 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. ...
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### Find area given two curves which belong to the orthogonal family to $x^2+2y^2=K$

Calculate the region area of the $xy$-plane limited by curves $C_1$ and $C_2$ with $|x|\leq4$, knowing that $C_1$ passes through $(1, 1)$, $C_2$ passes through $(1, -1)$ and both curves belong to the ...
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### How to calculate part of sphere area bounded by two planes?

Let we have two planes $p_1$ and $p_2$ that intersect by an angle $\beta$. Let $p_1$ be on distance of $a$ from the center $C$ of sphere with radius $1$ and $a<1$. Let intersection of the two ...