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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.

6
votes
1answer
321 views

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:=\...
0
votes
2answers
109 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. ...
1
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0answers
45 views

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 ...
0
votes
1answer
17 views

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 ...
0
votes
4answers
24 views

Does angle cause the change of area? [on hold]

Suppose there is a square, length =5cm. If one of its angle increases or decreases then it will be rhombus. But will it cause the area to change?
1
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2answers
32 views

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}$ ....
3
votes
1answer
78 views

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 ...
1
vote
0answers
45 views

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.
3
votes
2answers
80 views

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 ...
3
votes
2answers
105 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 ...
1
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0answers
27 views

Can I say that Integration can be equal to the formula of finding the area of a right triangle?

Let us have a certain function like $f(x)=x$ In the world of integration, can I say that $\int_{0}^bxdx=\frac{1}{2}(b)(f(b))$? Because if you look at graph of function x, it would look like something ...
-1
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2answers
34 views

Is it possible to cover all circle area with infinite lines starting from the center? [closed]

Is it possible to cover all area of a circle of radius r>0 with infinite lines starting from the center?
-2
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1answer
12 views

Area with centroids [closed]

enter image description here If the area of ACB is 60.5, what is the area of CXB?
5
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2answers
239 views

Grazing area for a goat around a circle.

I am doing this math question and i am really confused on how to approach it. This is the question: A retired mathematics professor has decided to raise a goat. He owns a silo and a barn. The barns ...
2
votes
0answers
40 views

Surface area of melting ice block

A cubic block of ice is melting and retains its cubic shape as it melts. Its volume (in $\rm m^3$) at time $t$ is given by $$ V = 4000-2000 \cdot e^{0.01t}$$ As the ice melts, the bottom ...
1
vote
2answers
4k views

Calculate area of triangle in space given points

Problem 5. a) Find the area of the space triangle with vertices $P_0, P_1, P_2$: $$ P_0 = (2,1,0),\ \ P_1=(1,0,1),\ \ P_2=(2,-1,1) $$ My current solution is to use $\frac{1}{2}\Big|(P_1-P_0)×(P_2-...
0
votes
1answer
18 views

Area of the common region of three circles.

Find the area of the region that is common to the circles $r = 1$, $r = 2 \cos θ$, and $r = 2 \sin θ$. I tried many ways to get the common region, but it seems impossible to eliminate to that point, ...
4
votes
3answers
54 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 ...
5
votes
2answers
118 views

Area of cut-out of three circles

Let three circles at different centers with different radii be given. They might intersect as shown in the picture. How to derive the area of the set (blue) in terms of the centers and radii? ...
1
vote
2answers
1k views

Area of one of four regions within a rectangle

There is a figure below (a rectangle). You can see different colors depicting different regions of the figure. The labels on the top of a region defines the area of that region. Can you find the ...
0
votes
0answers
23 views

Area under curve given a set $\{ [x, y] \in \mathbb{R^2} \ | \ 0 \le x \le 1 \ \& \ 0 \le y \le x \arctan^2 x\}$

Evaluate area under the curve for: $\{ [x, y] \in \mathbb{R^2} \ | \ 0 \le x \le 1 \ \& \ 0 \le y \le x \arctan^2 x\}$ I know that to find the area under a curve of a function from a to b, ...
0
votes
3answers
34 views

find area of kite given side length

Circle with two tangent lines Above is the picture in question. A circle is given, center (-2,4) and a point outside the circle (0,10) is shown. Asked to calculate the area of the quadrilateral ABCD, ...
0
votes
1answer
46 views

calculating the area in polar coördinates

I have difficulties calculating the area and setting the right boundaries of the following polar coördinates: $$r=2(1+cos(\theta) ) $$ Thanks in advance
1
vote
1answer
45 views

Find the area of a sphere inside a function

I want to find the area of the portion of the sphere $x^2+y^2+z^2=4z$ inside the function $x^2+y^2=z$ using double integrals. The graph would be something like this: Because of the nature of this ...
2
votes
1answer
82 views

Find the area limited by 4 curves, using change of variables

I'm trying to show that the area bound by the curves $r^2= 3\cos(2\theta)$, $r^2= 4\cos(2\theta)$, $r^2= 3\sin(2\theta)$, $r^2= 4\sin(2\theta)$ in the first quadrant is equal to $$A= \frac{10 - 7\...
5
votes
0answers
70 views

Is this result already a known theorem in geometry?

I have been playing around with geometry and I found that: Let two perpendicular lines intersect at a point that is inside a circle. Then the area of the quadrilateral formed by the vertices made by ...
18
votes
7answers
16k views

Show that the area of a triangle is given by this determinant

I'm not sure how to solve this problem. Can you guys provide some input/hints? Let $A=(x_1,y_1)$, $B=(x_2,y_2)$ and $C=(x_3,y_3)$ be three points in $\mathbb{R}^{2}$. Show that the area of $\...
3
votes
2answers
59 views

Finding area of triangle given ray of altitude

three altitude intersecting each other at H. Find the area of triangle ABC My attempt feel like cheating a little bit, I let BA = $\dot{\vec{C}}$, AC = $\dot{\vec{B}}$ and CB = $\dot{\vec{A}}$ then ...
8
votes
2answers
96 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
47 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 ...
-2
votes
5answers
182 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?
1
vote
1answer
24 views

Find area under $y= x^2 - x^4$ from x=-1 to x=0 using the Riemann sum

I'm trying to find the area under $y= x^2 - x^4$ from $x=-1$ to $x=0$ using the Riemann sum. This is what I've done so far: $\Delta x = 1/n$ $x_i =-1 + i/n$ $A = R_n = \lim_{n\to \infty} {\sum_{i=...
3
votes
4answers
130 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 ...
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
39 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
33 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 ...
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
0
votes
1answer
971 views

Double integral of off centre circle.

I have the vector field $F = (3xy,-x)$ along the circle $c$ (counter clockwise) which has a radius $a$ and centre $(a,0)$. I want to try and apply Green's Theorem to this, where I obtain $\int\int(-1 ...
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\...
-3
votes
3answers
73 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
137 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....
0
votes
1answer
855 views

Find area of region that is common to both squares.

A 3-meter square and a 4-meter square overlap as shown in the diagram.D is the center of the 3-meter square. Find the area of the region DGFE. I 've tried to form right triangles in such region but ...
0
votes
0answers
29 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
0answers
18 views

Calculating density of individuals within an area (tree stand density)

Any help greatly appreciated! I want to find the density of trees surrounding a sample point (measured in trees/m$^2$). The distance from the sample point to the three nearest trees has been ...
0
votes
1answer
43 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
22 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 ...
-1
votes
2answers
48 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 ...
0
votes
1answer
44 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
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 ...