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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197 views

How to find the area of the rectangular region 2 ≤ x ≤ 5, -1 ≤ y ≤ 3

How to find the area of the rectangular region $2 ≤ x ≤ 5$, $-1 ≤ y ≤ 3$. I tried to plot the graph in $xy$-plane, but I'm not sure how to find the area.
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75 views

How do I calculate?

How do I calculate cross-sectional area of this: ${r}=\sqrt{sin\theta} \, $, when $ 0 \le \theta \le \pi,$ Don't know what is the right answer but I have get that the area is 1. Is that right answer?...
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1answer
33 views

How do I calculate area?

How do I calculate the area of this: $D=\{ (x,y)\mid 0 \le x \le 1, x^2 \le y \le x^2+2 \}$ ${A}=\iint_D \, \textrm{d}A.$ Don't know what is the right answer but I have get that the area is 6. Is ...
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1answer
48 views

minimum area occur when $c=\frac{a}{2}$

Let $f(x)=\sqrt{\tan x}$ . Then show that the area bounded by $y=f(x),y=f(c),x=0$ and $x=a,0<c<a<90^\circ$ is minimum when $\displaystyle c=\frac{a}{2}.$ what i try enclosed area $$A=\int^{...
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Given that $[ABC]$ : Area of small circle = $\frac{3\sqrt3}{4}$ : $\pi$. How many parts of area of small circle is inscribed in large circle? [closed]

In the common region of two circle, $\triangle ABC$ has been drawn with its maximum area such that the proportion of the maximum area of $\triangle ABC$ and the area of small circle is equal to $\frac{...
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3answers
63 views

What is the area of $ABCD$ parallelogram where $E$ is mid-point of BC and the area of $BEC$ is 126?

$ABCD$ is a parallelogram. Point $E$ divides $BC$ into two equal lengths. If the area of $BEF$ is 126, what is the area of $ABCD$? Source: Bangladesh Math Olympiad 2017 Junior Category. I can not ...
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341 views

Find the area of the shaded region of two circle with the radius of $r_1$ and $r_2$

In the given figure , $O$ is the center of the circle and $r_1 =7cm$,$r_2=14cm,$ $\angle AOC =40^{\circ}$. Find the area of the shaded region My attempt: Area of shaded region $=\pi r^2_2 - \pi ...
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1answer
90 views

Area of a region delimited by chords and circular arcs.

Let $AB$ be the diameter of circle $O$, where $AB = 2$. Circle $P$ is internally tangent to circle $O$ at point $B$, and $PB$ = $\frac{2}{3}$. Two different chords $AX$ and $AY$ are drawn tangent to ...
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Least Area bounded

Area bounded by f(x) = $x^3/3$ - $x^2$ +a and the straight line x=0, x=2 and the x axis is minimum. Find a This is the screen shot of the function when a=0.6. Scrolling the value of "a" from 0 to 9,...
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Find the maximum value of $\square OXPY$

Problem: There are moving point $X$ and $Y$ lie on the $x$ and $y$ axes, respectively. For moving point $P$, $PX=3$ and $PY=4$. Find the maximum area of $\square OXPY$.($O$ is origin). My solution: ...
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1answer
37 views

Surface area of spherical section delineated by 2 perpendicular circular planes/central angles

The problem concerns visible area based on a field of view from the center of a sphere. I was never taught spherical trigonometry so even basic terminology is hard. After trying to figure out the ...
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2answers
62 views

Area of Rectangle and circle

Just a basic question : The area of a rectangle is length×width because I'm filling the length with width. This makes sense to me but if I do the same approach for a circle then the area becomes $2\...
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2answers
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What is the area formed by the loop of the curve $x^3 - y^3= xy$?

I know how to find area of elementary functions by integration. The only loop questions that I know how to solve are the ones that are of the petal type, which can be converted into polar form and ...
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1answer
62 views

Area of sector bounded by line and curve

I have a line segment $\overline{AB}$, with endpoints $(x_1,y_1)$ and $(x_2,y_2)$. Drawing any function $f(x)$through those 2 points, the function must have an average velocity over $[x_1,x_2]$ of $\...
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1answer
49 views

Determine the area bounded by the curves $y=2x$ and $y=x^2$ and the parabaloid $z=x^2+y^2$

Determine the area bounded by the curves $y=2x$ and $y=x^2$ and the parabaloid $z=x^2+y^2$ I'm not really sure what my function that I'm integrating is supposed to be I believe I want to integrate on ...
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61 views

How can I determine the area of the region bounded by the curve $x^2=y^4(1-y^3)$ [closed]

$x^2=y^4(1-y^3)$ On a graph it looks like a diamond. This is for my integral calculus class.
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109 views

Bounded Area between $y=x^2$, and $y=9$, and $y=k$

I have been given a function to find the horizontal line $y=k$ which would splice the area bounded between $y=x^2$, and $y=9$, into two equal parts. I approached the function using symmetry to find ...
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1answer
16 views

How to tell the run of a line segment where the area underneath it, starting point, and slope are known.

I have a program where I am currently solving for the difference of X₁ and X₂ of a line-segment where the area under the line-segment (A), the slope (s), and the starting point (X₁,Y₁) are known. ...
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14 views

Surface area of ellipsoid created by rotation of parametric curve

I have a parametric curve (elipse) defined as follows $$\begin{aligned} x(t) &= \cos(t)\\ y(t) &= 2 \sin(t)\end{aligned}$$ and need to calculate the surface area of the ellipsoid produced by ...
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51 views

Solid of revolution using washer method(Gives negative answer)

My teacher even knew that the answer should not be negative but it turned out to be negative. The given was y=x^2, y=4x-x^2, revolving about the y-axis. Here are some of the solution presented, I hope ...
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1answer
56 views

Area of the surface

Compute the area between the surfaces $(x^2+y^2)^2=a^2(x^2-y^2)$ and $x^2+y^2+z^2=a^2.$ My attempt: I take the first case $z\geq 0$: $$x=x$$ $$y=y$$ $$z=\sqrt{a^2-(x^2+y^2)}$$ then $$T_x=(1,0,\...
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1answer
53 views

Finding enclosed area

Hints only please !! I gave a test today, and it was asked to find area enclosed by the curve $x^4 + y^4 = 2*x*y$. This is an implicit function. A quick obs. shows me that, the curve is entirely bound ...
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3answers
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Under between the x- and y-axis of the graph of $y=x^n$

Consider the graph of $y=x^n$ for $n>1$ and $x>1$. The area bound between the curve and the x-axis between $1$ and $a$ is one third the area between the curve and the y-axis between the values ...
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0answers
37 views

How to formally prove this integral identity?

I was doing some investiations into 'areas under curves' an found by a geometric analogy the following integral identity: $$ \int_0^a f(x)dx =\frac12 \int_0^{f ′ (a)} (a- f^{-1}(f′(x)))^2 dx $$ ...
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2answers
70 views

Finding out the value of $\angle DQC$ in a trapezium $ABQD$ where $\angle DCB$ = 30$^\circ$

In this below diagram, $\angle ABC=60^\circ, \angle DCB=30^\circ $, $AD$ is parallel to $BC$ and $AP$ is perpendicular to $BC$. Both the area and perimeter of $ABCD$ and $APQD$ are equal . What is the ...
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42 views

Improper integral - Calculating Area of Floor Function

I'm stuck trying to calculate the area of the region defined by: $\begin{Bmatrix} {(x,y)\in\mathbb{R^2}:0\leq x\wedge 0\leq y\leq 2^{-\lfloor x \rfloor}} \end{Bmatrix}$ I'm just starting studying ...
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Area of a right trapezoid with bases $15$ and $8$, and height $9$

I can't figure out how they got that. I got 1080, they got 103.5. How???
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22 views

Please explain how to take limits in double integral while finding volume using the given problem

Question: Find the volume under the surface $z=\sqrt{1-x^2}$ and above the triangle formed by $y=x$, $x=1$ and $x$ axis. The two integrals are given as follows: $$\int_0^1 \int_y^1 \sqrt {1-x^2} \,...
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Generalizing computation of number of pixels in inscribed circle to axis-aligned ellipses at arbitrary points

This answer really nicely sums up the question of how to compute the number of pixels inside an inscribed circle. However, I am looking for a more generalized version of this in two ways. 1) I would ...
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2answers
75 views

Proof for area of an equilateral triangle with respect to one side?

I'm trying to find out the area of an equilateral triangle with respect to one side. Anything wrong with my proof? An equilateral triangle with sides of length $a$ can be divided in half along the ...
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1answer
59 views

Inequalities about area and perimeter

"A gardener is laying out a rectangular lawn. His specifications are that the area $(A)$ must be greater than $40$cm but the perimeter $(P)$ must be less than $40$cm. if the width of the lawn $(w)$ ...
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1answer
33 views

Help indentifying unknown method of computing area by rolling a ball about the perimeter

Some time ago I saw a brief presentation about a newly discovered method used to compute the area of certain two-dimensional shapes by rolling a circle (of varying radius) about the perimeter; the ...
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1answer
29 views

Is this the simplest way to visually prove the “scalene trapezoid area formula”

Please refer to image. Is there a simpler way to visually prove the scalene trapezoid area formula?
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1answer
55 views

Find area of the fourth triangle given the area of three triangles.

This is the question that I got in TCS Ninja under the Quantitative section. How shall I do this? Help !!
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1answer
58 views

Area of a circle segment on sphere, given radius (meters) and central angle (degrees)

Situation I have a circle segment and some information about the circle it belongs to. Given Information: radius of the circle in meters central angle in degrees lat/long of all three points on the ...
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1answer
85 views

How to double the circle?

I'm looking for a compass-and-straightedge method to construct a circle that has area twice of the area of another circle, with no prior knowledge of π, without knowledge of the formula for the area ...
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51 views

How do I find the area bounded by a $y=\frac{4}{3}x^2+\frac{12}{3}x-3$ and $y=\sqrt{x}$

I am having difficulties with this problem:$y=\frac{4}{3}x^2+\frac{12}{3}x-3$ and $y=\sqrt{x}$. Graphng two of the functions I get the following: Graphing both functions shows that they intersect at $...
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how to calculate wall thickness of a mesh?

based on similar questions on mesh volume, volume of a mesh can be calculated by following equation: volume = ((vec1 x vec2) . vec3) /6 where vec1, vec2, and vec3 are the vectors from origin to a ...
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1answer
129 views

Determining the area of a right triangle, perimeter given, hypotenuse value given in terms of one of the legs.

The problem states: Right Triangle- perimeter of $84$, and the hypotenuse is $2$ greater than the other leg. Find the area of this triangle. I have tried different methods of solving this ...
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1answer
33 views

Find region of $0\le y\le1$ and $y\le x\le1$ [closed]

How i can find the region that is bouned from these inequalities? Any general rule when we have to deal with these inequalities? Ploting the graphs?
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2answers
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Function whose graph is dense in the plane [duplicate]

Is there a function $f:\mathbb R\to\mathbb R$ such that for every disc in $\mathbb R^2$ the graph of that function has at least one point that lies inside that disc? I searched for something similar ...
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1answer
165 views

Help find work done to pump fluid

A tank is full of oil weighing 20 lb/ft${^3}$. The tank is an inverted right rectangular pyramid (with the base at the top) with a width of 2 feet, a depth of 2 feet, and a height of 5 feet. Find the ...
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2answers
224 views

Orthogonal projection of an ellipsoid [closed]

Suppose an ellipsoid given by $\{{(x,y,z)| {x^{2}+\frac{y^{2}}{4}+\frac{z^{2}}{9}}}=1\}$, find the area of the orthogonal projection of the ellipsoid on the plane ${2x+4y-5z=10}$. What is the right ...
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148 views

How to determine the area of a rotated ellipse?

The ellipse $6x^2+4xy+5y^2+8x+8y+1=0$ is neither expressed in terms of $x$; like $y=\pm\sqrt{a^2-x^2}$, nor in terms of $y$; like $x=\pm\sqrt{a^2-y^2}$. Separation of $x$ (or $y%$) may be impossible. ...
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80 views

How to calculate the area and volume of a random 3d shape knowing only its coordinates?

Given a set of 3d points which make up an vector object of any shape (with any number of points), without the edges being known, how can the object's edges be found/detected so that the object's ...
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2answers
35 views

Area that is bounded by functions

The functions $$f_k(x)=\frac{x+k}{e^x}$$ are given. Let $A(u)$ be the area that is bounded by $f_1, f_3$, the $x$-axis und the line $x=u$. I want to check the area if $u\rightarrow \infty$. $$$$ ...
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1answer
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Calculate Overlapping Area of $2$-Dimensional Shapes

I am running a Computer Simulation where 2 Shapes are moving towards each other and will eventually overlap. I want to calculate the overlapping Area of the shapes - in this example a Circle and a ...
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2answers
92 views

Find area of quadrilateral in triangle. [closed]

What is the area of $HIJK$ quadrilateral, if the area of $ABC$ triangle is $70$, $BE=ED=DA$, and $BF=FG= GC$?
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3answers
63 views

Finding percentage increase of area [closed]

Find the percentage increase in the area of a triangle if its each side is doubled?since no information is given about the type of triangle in question, so should i take a equilateral or isosceles or ...
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2answers
96 views

Do the given perimeter and area corresponds to many shapes? [closed]

I have a perimeter P and area A of a planar shape. How to prove that there are many shapes that corresponds to those perimeter and area values?