Questions tagged [area]
Area is a quantity that expresses the extent of a two-dimensional or three-dimensional surface or shape.
2,354
questions
1
vote
1answer
48 views
Maximization problems
I am in a introductory course of calculus in several variables and i have these problems.
A farmer wants to build a corral with pentagonal form(not regular) that is formed by the union of a rectangle ...
1
vote
0answers
17 views
using partial derivations to get the approximate error for this problem
The area of a triangle is $A=\frac{1}{2}a*b*sin(c)$
$𝑎*𝑏* sin( 𝑐)$, where $a$, $b$ are two sides of the triangle and $c$ is
the included angle. In surveying a triangular plot of land, $a$ and $b$ ...
0
votes
3answers
59 views
Estimating Area Exam Question
Pieces of turf are 1m long by 0.5m wide. Each piece costs £3.79 .
$1 * 0.5 = 0.5\text m^2$
a)Estimate the cost of turf required to cover these spaces.
i) 9.6m by 2.4m
$10 * 2 = 20\text m^2$
...
0
votes
1answer
36 views
Calculate area enclosed by 4 curves
I am trying to find the area enclosed by 4 piecewise smooth curves. As can be seen from the figure,
BLACK curve is a segment of a circle, C
...
3
votes
4answers
169 views
Definite integral of $1/(2\sin^4x + 3\cos^2x)$
I have $f = \frac 1 {(2\sin^4x + 3\cos^2x)}$ which area should be calculated from $0$ to $\frac{3\pi}2 $.
I noticed that $$\int_0^{\frac{3\pi}2} f \,dx= 3\int_0^{\frac{\pi}2} f \,dx$$
I tried to ...
8
votes
2answers
97 views
Calculating how much light gets through steel mesh (commonly used to make cages)
I have expanded steel mesh that I use to make garden cages:
I would like to know how much sunlight the mesh lets through.
I think I need to calculate the area of the mesh's negative space. And then ...
1
vote
1answer
36 views
find the area enclosed by $ f(x)=x+\sin(x)$ and its inverse from $x=0$ to $x=2$ [closed]
I don't have a single clue to start, and we cant find the inverse so we must use some properties, but which ones?
thanks
-3
votes
3answers
82 views
I Need Help in a Challenge [closed]
My teacher challenged me with the question below:
$$\sqrt{\frac{\sqrt{41}+\sqrt{29}+\sqrt{10}}{2}\ast \left ( \frac{\sqrt{41}+\sqrt{29}+\sqrt{10}}{2} - \frac{\sqrt{41}}{1} \right )\ast \left ( \frac{\...
7
votes
2answers
144 views
How can i Prove that the gray area is the same as white area? [duplicate]
A circle is cut into 8 parts, each part has the angle 45 degrees from an arbitrary point. how to prove that the white area is the same as the Gray area?
I just want any hint for solving this question....
3
votes
1answer
82 views
Find area of cylinder $x^2 + y^2 = r^2$ that satisfies $0 \le z \le y$
I think I can imagine the shape of surface area. This is what I did:
$$\begin{eqnarray*}
\text{Surface area} & = & \int_{0}^{\pi} r y \sqrt {2} \, d\theta \\
& = & \int_{0}^{\pi} r\...
0
votes
0answers
37 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 ...
0
votes
1answer
45 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).
1
vote
1answer
52 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
votes
3answers
46 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 ...
0
votes
1answer
52 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}...
0
votes
0answers
25 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 ...
0
votes
4answers
54 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
$$\pi r^2 \frac{\theta }{360} \space \space \text{ (degrees case)}$$
$$or$$
$$r^2 \frac{\theta }{2} \...
0
votes
1answer
64 views
Area bound by $y =\ln x +\tan^{-1} x$?
I'm not really sure how to do this without having a real graph using a graphing calculator. Adding the integrals would no work since some areas are overlapping, and even subtracting doesn't since one ...
1
vote
1answer
46 views
Finding $\iiint 6z\,dx\,dy\,dz$ over $\lbrace (x,y,z) \in \mathbb{R}^3 : |x+y| \le z \le |x|+|y|\le 1 \rbrace$
I want to calculate integral : $\iiint 6z\,dx\,dy\,dz$. The area is $$\Omega= \lbrace (x,y,z) \in \mathbb{R}^3 : |x+y| \le z \le |x|+|y|\le 1 \rbrace$$
My problem is this, that I don't know what is ...
0
votes
1answer
32 views
Determine the area enclosed by the curve
Determine the area enclosed by the curve with two polar equations:
$r= \sqrt 2 \sin(\alpha)$
$r^2 = \sin(2\alpha)$
I have no clue how to do this. A formula we are given is the one below but I'm ...
2
votes
1answer
47 views
Number of Atoms on Earth's Surface (Question based on Vsauce video which involves fractal dimension)
In a Vsauce video titled "How much of the Earth can you see at once" they try to do a calculation to estimate the number of atoms on Earth's surface.
The first part is easy:
1) Calculate surface ...
3
votes
1answer
68 views
heron's formula proof
I have seen an interesting proof of Heron's formula here. It is very simple, but I do not understand one point. The author demands, that the formula should contain factor $(a+b+c)$, because when we ...
0
votes
0answers
21 views
Determining area of graph based on unknown boundaries bounded by special trigonometry graphs
Interesting question for all to solve that i chanced upon luckily.
let the area bounded by two graphs below from $m \pi $ to $(n-1) \pi $ be A, where $m$ and $n$ are integers. Express $n-m$ in terms ...
2
votes
0answers
21 views
Find sub areas of a function in a circle
I have a cellular signal calculation function, which calculates the signal given the distance from the antenna. Without the constants, the function is basically: $f(d)=1/(d^α)$ where α is a parameter.
...
1
vote
1answer
71 views
Evaluating Area Under a Curve [closed]
Consider the curves:
$$\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}$$
Where $a$,$b$,$c$,$d$,$k$ $...
1
vote
1answer
50 views
Area inside $r = 1 + \cos \alpha $
We are studying polar equations.
Calculate the surface inside $r = 1 + \cos \alpha $ and outside $r = 1$.
I know the area inside $r = 1 + \cos \alpha $ being $\frac{3 \pi}2$ because I calculated $\...
1
vote
3answers
93 views
What is the area of a triangle with sides $\sqrt{5}$, $\sqrt{10}$, $\sqrt{13}$?
I found a "fun algebra problem" that asks you to find the area of a triangle whose sides are $\sqrt{5}$, $\sqrt{10}$, $\sqrt{13}$. After some algebra hell trying to work with Heron's formula, I ...
-1
votes
2answers
56 views
A given perimeter length that is circular encloses the maximum area - which are the (analytic) proofs? [duplicate]
I'm guessing Newton, because of his integrals. But what proofs have been established, and which is the most mathematically intuitive one?
I was looking for the tag "circumference", supplied the newer ...
3
votes
4answers
89 views
Finding the area between two curves.
Context: High School question.
Find the surface area between the curve of the function $y=6-3x^{2}$
and the function $y=3x$ in the interval $[0,2]$
My approach:
-We must find the points of ...
0
votes
0answers
17 views
Proving an integration identity using the co-area formula
I want to prove that identity: $$\int _{B^n(0,R)} f(x)dx = \int _0^R r^{n-1} \int _{S^{n-1}} f(ry)dS(y)dr$$
where $B^n(0,R)$ is $n$-dimensional ball and $S^{n-1}$ is the $(n-1)$-dimensional sphere (...
0
votes
1answer
32 views
Maximizing the size of n similar equilateral triangles from a rectangle
I have 3 rectangles of greenhouse sheeting material. They are each 12 feet long and 4 feet high.
I want to use this material to clad a 3/4 icosahedron dome structure. What that means is that I will ...
1
vote
3answers
62 views
Solve $\int^2_{-1} (1-x)dx$ by thinking in terms of area?
On our last quiz in Calculus 1, my professor asked us to solve $\int^2_{-1} (1-x)dx$ by thinking in terms of area. I don't know what he means by, "thinking in terms of area". I can solve it myself, ...
0
votes
0answers
32 views
Polar area and do I need absolute value signs?
Imagine a curve such as
$$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}\...
0
votes
0answers
29 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
43 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 ...
0
votes
1answer
34 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 ...
7
votes
2answers
105 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
votes
0answers
33 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^{\...
0
votes
1answer
30 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 ...
0
votes
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 ...
1
vote
3answers
36 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 ...
0
votes
1answer
26 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
(...
0
votes
1answer
81 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:
$$[\frac{x^2}{...
0
votes
1answer
34 views
Given a set of points, find maximum area of triangle
Given a finite set of 2-d points, I need to find the maximum area of triangle formed.
My solution steps :
Take mean of points , lets call it (x_m, y_m).
Take 3 most distant points from (x_m, y_m).
...
5
votes
2answers
67 views
Area below the curve $y=\left[\sqrt{2+2\cos2x}\right]$
Find the area below the curve $y=[\sqrt{2+2\cos2x}]$ and above the $x$-axis , $x\in [-3\pi,6\pi]$, (where $[.]$ denotes the greatest integer function) .
My approach:
$$y=[\sqrt{2+2\cos2x}]$$
$$=[\...
4
votes
1answer
116 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=...
0
votes
0answers
46 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, ...
1
vote
1answer
40 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 ...
0
votes
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 ...
-2
votes
5answers
202 views
How can I find the area of the shaded region given by two semicircle and a line segment? [closed]
I don't know how to start a problem of this type. How can I find the area of the shaded region?
