Questions tagged [area]

Area is a quantity that expresses the extent of a two-dimensional measurement of a shape.

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4answers
38 views

Find the area of region between the $x$-axis and the graph of $f(x)=x^3-x^2-2x$, for $-1\le x\le2$

The final answer I have from calculation is $37/12$ sq. units. The final answer was in negative, but because area can't be negative so I solved it making positive. Can someone help me in getting the ...
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3answers
75 views

Area bounded by $2 \leq|x+3 y|+|x-y| \leq 4$

Find the area of the region bounded by $$2 \leq|x+3 y|+|x-y| \leq 4$$ I tried taking four cases which are: $$x+3y \geq 0, x-y \geq 0$$ $$x+3y \geq 0, x-y \leq 0$$ $$x+3y \leq 0, x-y \geq 0$$ $$x+3y \...
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1answer
43 views

How to solve the problems in which wire is cut into 2 parts

A wire of length 2 units is cut into two parts which are bent respectively to form a square of side $=x$ units and a circle of radius $=r$ units. If the sum of the areas of the square and the circle ...
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0answers
9 views

Closed Line integral of a scalar field

If $f(\mathbf r)$ is a scalar field, then line integral of this function is a cross-sectional area (the shaded part in the picture) bounded by $f(\mathbf r)$ (in blue) & curve $C$ ( curve is in ...
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1answer
14 views

Finding the Area of a Piecewise Function

I'm learning applications of integration and this is my first experience with this sort of question. How would I get started on a problem like this?
4
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1answer
65 views

Find the area of this pentagon

Let $BCDK$ be a convex quadrilateral with $BC=BK$ and $DC=DK$. $A$ and $E$ are points such that $AB=BC$, $DE=DC$ and such that $ABCDE$ is a convex pentagon. Point $K$ lies in the interior of pentagon $...
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2answers
35 views

How to show that $A$ is increasing?

Suppose that $f$ is a twice differentiable real function such that $f''(x)>0$ for all $x\in[a,b]$. Find all numbers $c\in[a,b]$ at which the area between the graph $y=f(x)$, the tangent to the ...
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2answers
30 views

Find $\int_{\left(C\right)}xy{\rm d}x+y^{2}{\rm d}y$ with $\left(C\right)$ bound by $y\geq 0,x^{2}+y^{2}=4\left({\rm clockwise}\right).$

Prob. Find $\int_{\left ( C \right )}xy{\rm d}x+ y^{2}{\rm d}y$ with $\left ( C \right )$ closed by the path $y\geq 0, x^{2}+ y^{2}= 4\left ( {\rm clockwise} \right ).$ My attempt: $\int_{\left ( C \...
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1answer
36 views

Find the slope of the line go through a point such that the area between the graph and the line is minimum

If $a$ is the slope of $(L)$ that go through point $(-1,2)$ and $f(x)=x^2$ then find $a$ such that the area between the graph and the line is minimum. For what I thought, the area is minimum if the ...
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0answers
20 views

How to compute the area of a $k$ dimensional sub-manifold in a $n$ dimensional manifold? [closed]

How to compute the area of a $k$ dimensional sub-manifold in a $n$ dimensional manifold? Given a manifold $(M,g)$, where $g$ is the Riemannian metric. If $S$ is a $k$-dimensional sub-manifold of $M$, ...
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2answers
23 views

parametrization of surfaces and area

The question is: Paraboloid $z=x^2+y^2 $ divides the sphere $x^2+y^2+z^2=1$ into two parts, calculate the area of each of these surfaces. I know that i need to use $\iint |n| dS$ but i need to first ...
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1answer
32 views

Finding area of quadrilateral inscribed inside of a semicircle

I've been practicing problems recently to study for the upcoming AMC10, and came across one I could not figure out how to solve one of them. The diagram of it is attached below, and the problem goes a ...
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1answer
18 views

Hunt for trapezoid area & arithmetic progression sum formula similarities.

When I was looking into Arithmetic progression sum formula, I found out that it is similar to Trapezoid are formula. Arithmetic ...
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2answers
59 views

Partition the remaining rectangle into equal parts.

There’s a rectangle. A small rectangle is cut from the bigger rectangle ( not necessarily from the center). How will you partition the original rectangle after removing the cut such that the remaining ...
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0answers
25 views

Area of a square with an arc passing through it

Assuming the the blacks and blue arcs are quarter circles, how would I go about finding the region A+B? Region A was pretty easy to find, as it's just the area of a circle with radius $r$ divided by $...
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1answer
30 views

Using jacobian to transform area elements from one system to another.

Please excuse my lack of rigor,I'm just average Physics undergraduate. I have a transformation from $(u, v)$ to $(x, y)$. So an infinitesimal area element from $ (u, v)$ to $(x, y)$ plane will ...
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1answer
76 views

Finding the ellipsoid of maximum area inside a square - and extension to $\mathbb{R}^n$

In an answer here, I read that: Take a max area ellipse. Apply an affine transform to make it a circle; then the problem becomes to show that a minimal area parallelogram containing a circle is a ...
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1answer
44 views

Area of metric ball on n-sphere

Suppose $S_n = \{x \in \mathbb{R}^{n + 1} : ||x||_2 = 1\}$ is the $n$-sphere. Let $d : S_n^2 \to \mathbb{R}$ be the angle metric on $S_n$, i.e. $d(x, y) = \arccos(x \cdot y)$, where $\cdot$ is the dot ...
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1answer
51 views

A question about the polar equation $r=ln(\theta)$

Take the polar equation r=ln(theta) for theta between $0$ and $2\pi$. 1 In this, we have a loop that connects at a point $(x,y)$ for $2$ unique points $(r_1,\theta_1)$ and ($r_2,\theta_2)$. There are $...
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2answers
53 views

Calculate area of a circle

\begin{align} π &= 3.1415\dotsc\\ s&= πr^2 \end{align} Because $π$ infinitely continues, does it means $s$ is not ever a right answer, does it mean we do not know exact $s$ of a circle? Is ...
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2answers
45 views

How to calculate the Area of a “Polygon” including Arcs

I have a set of points forming a polygon. However, any 3 points in this polygon can also be represented as an arc (starting at point 1, through point 2, to point 3). I need to find the area of this ...
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1answer
32 views

How to find area between curves? ($\frac{1}{x}$ problem)

I need to find the area between those curves: $f(x)=\frac{1}{x}$, $f(x)=6e^x$, $f(x)=1$, $f(x)=6$ I calculated the limits of integration, it is $\ln(\frac{1}{6})$ to $1$. But we cannot integrate ...
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0answers
24 views

Calculating areas in a square

Enclosed within a square of side A is another smaller square of side B. The center C of side square B is off-centered with coordinates (p,q); OC makes $\alpha$ to x-axis. How to find areas enclosed ...
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1answer
44 views

Analytical approach to find area of parallelogram

I know that to find the area of parallelogram, we have to either find the cross product of its adjacent sides or half the cross product of its diagonals. However I've encountered a question to find ...
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5answers
86 views

Area of square inside a triangle [closed]

Considering the attached image, please compute the area of the square. $\\$To elaborate suppose we have a triangle $\Delta ABC$, that its angle $\angle ABC$ is equal to $45^o$. On side $\overline{AB}$ ...
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1answer
61 views

How would I go about solving for the grayed out area?

I was asked to get the grayed out area that you see in this image. All that I'm given to solve this is the three points of the main triangle (A, B, C) and the center of the circle (X) Also the dashed ...
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1answer
50 views

Area between parabolas

I would like to calculate the area between these 2 curves. As you can see the first one is $x^2 = αy$ and the second one is $y^2 = 2αx$ What I tried is to solve the equation when the two curves are ...
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0answers
5 views

Auto Arrangement and Positioning of nested containers

We are trying to solve a problem of auto-layout or auto-arrangements of containers. The containers can be in a hierarchy and can be defined by the rules below: All shapes are containers - Rectangle ...
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1answer
67 views

How would one approximate the surface area of a curved shape as on these ETFE pillows?

I need to get the surface area of these ETFE pillows:
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2answers
53 views

Illusion in maths?? (Area of sectors and segments) [closed]

I was practising for my maths Olympiad when I stumbled upon this question... This question is more than just a figure. It actually isn't what it looks like. I could've simply seen the solution at the ...
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1answer
30 views

Surface Area of Cap below, Top Part of Sphere

How do I find the Surface Area of an oval cap below (Top of sphere)? Oval Cape Internet is stating: 2πRh However, a simple 8 radius circle is ...
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2answers
54 views

Calculating the area under the curve $e^x$ without calculus

I was thinking about the integral $$\int_{0}^{1}e^xdx$$ This is a problem that can be solved in seconds using calculus: the answer is $e-1$. But, I was wondering, is there a way of solving this ...
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4answers
100 views

Area of triangle with incircle tangent to semicircle whose diameter is on one triangle side

Triangle $\triangle ABC$ with incenter $I$ and inradius $r$ has the following property: the incircle is tangent to the semicircle with diameter $AB$ and is within the semicircle. Find the area of $\...
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1answer
37 views

Line integral using substitution

Using Green formula calculate the area that is bounded with curve:$(x^2+y^2)^2=2a^2(x^2+y^2)$. My main problem is how to find $x,y$ in polar coordinates. When i worked with double integrals i would ...
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0answers
24 views

Optimising surface area of a styrofoam cup

I am trying to optimise the surface area of a normal styrofoam cup while keeping volume constrained. When I tried to find its volume and surface-area equations online, I am finding that it involves ...
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0answers
28 views

Naming areas adjacent to a geometric shape

I'm writing documentation for a software I am working on, and I need some help with naming the areas that surround a shape relative to its acceptable travel path. Let's assume there is a blue ...
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2answers
63 views

find the greatest area of a parallelogram of two vertices lie on $x/a+y/b=1$ and the other two vertices lie on the two axes [closed]

find the greatest area of a parallelogram of two vertices lie on $x/a+y/b=1$ and the other two vertices lie on the two axes
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0answers
58 views

What is the area enclosed within $y=\frac{1}{x}$ and its axes of symmetry?

What is the area (exact and approximate value with steps would be appreciated) bounded by the horizontal and vertical asymptotes $(y=0, x=0)$ and the graph of $y=\frac{1}{x}$? i.e. $2$[(area under $y=\...
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1answer
64 views

Area between paraboloid cut out by cylinder

Find area between surface $z^2=x^2+y^2$ and $x^2+y^2=2x$. So, after using polar coordinates $x=r\cos \phi,y=r\sin \phi$ i get $0\le r \le 2\cos\phi$ and $0\le \phi \le \frac{\pi}{2}$. Than i plugged ...
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0answers
21 views

Double integral over a not-aligned square

I am studying multivariable calculus and came across a question asking me to evaluate a double integral of F(x,y) over a square, but with vertices at (0,0),(2,1),(3,-1), and (1,-2). Clearly the reigon ...
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0answers
43 views

Total average curvature of a surface?

Given a surface $S$ at any point $P$ of $S$ 2 curvatures are defined, the Gaussian curvature, which is the product of the principal curvatures, and the average curvature, which is their arithmetic ...
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1answer
82 views

What is the basic idea behind calculation of area? [closed]

The system of calculating area in terms of square units is pretty philosophical and not very intuitive. It must have taken a great amount of time for humanity to arrive at such a convention and to ...
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2answers
34 views

Find the minimum value of the ratio $\frac{S_1}{S_2}$ using given data

3 points $O(0, 0) , P(a, a2 ) , Q(b, b2 )$ are on the parabola $y=x^2$. Let S1 be the area bounded by the line PQ and the parabola and let S2 be the area of the triangle OPQ, then find min of $\frac{...
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1answer
27 views

If a curve $y=a\sqrt x +bx$ passes through $(1,2)$ and the area bounded by the curve, line $x=4$ and the $x$ axis

The way solve this is by integrating the function from 0 to 4 ie $$\int_0^4 f(x) =8$$ since the given curve clearly intersects $(0,0)$ But if try to find a point where $y=0$ by setting $y=0$ Then $$0= ...
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0answers
79 views

What is the average projected area of a thickened non-convex body?

The average projected area theorem states that for a convex body in 3D the average projected area $\langle \sigma\rangle$ for a random orientation is 1/4 of the surface area $A$ (despite the PDF often ...
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0answers
21 views

Summing squares within an irregular polygon

I am trying to know how many rectangles with a constant base and height (3mx1.8m) can fit inside an irregular polygon. To solve this, I'm taking the irregular polygon and putting small squares (0.6mx0....
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0answers
21 views

Find the surface area of the region defined by the intersection of $z=2y$ and $z=x^2+y^2$

I know A(s) = $\iint\sqrt{1+\frac {dz}{dx}^2+\frac {dz}{dy}^2}dA$ and solving for the domain D I can get to $x^2+(y-1)^2=1$. So logically, my answer should be $\int_0^2\int_{-\sqrt{1-{(y-1)^2}}}^{\...
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2answers
73 views

Finding the area not by integral

Original problem. Find the area under $y= \sqrt{x}$ in the range $\left [ 0, 1 \right ]$ My friend, she wants to use total area instead of calculating the integeral. So I tried something: Dividing ...
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1answer
49 views

Two goats on a sliding leash in a square garden

I have two goat problems I am trying to figure out. Here are my problems. You have a square with sides of length 100 m. There are two goats and each goat is on their own diagonal on a 10 m leash that ...
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2answers
42 views

Determine the area of the ellipse $A=\{(x,y) \in \mathbf{R}^² \mid 3x^2+4y^2\leqslant12 \}$

Determine the area of the ellipse $A=\{(x,y) \in \mathbf{R}^² \mid 3x^2+4y^2\leqslant12 \}$ I tried to use polar coordinates here, but couldn't get it to work. If I have $x=r\cos(\theta)$ and $y=r\...

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