Questions tagged [area]
Area is a quantity that expresses the extent of a two-dimensional measurement of a shape.
2,961
questions
-2
votes
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 ...
4
votes
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 \...
1
vote
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 ...
0
votes
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 ...
0
votes
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
votes
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 $...
0
votes
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 ...
2
votes
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 \...
0
votes
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 ...
0
votes
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$,
...
0
votes
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 ...
0
votes
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 ...
1
vote
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 ...
2
votes
2answers
59 views
Partition the remaining rectangle into equal parts.
Thereās 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 ...
0
votes
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 $...
0
votes
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 ...
1
vote
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 ...
1
vote
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 ...
0
votes
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 $...
0
votes
2answers
53 views
Calculate area of a circle
\begin{align}
Ļ &= 3.1415\dotsc\\
s&= Ļr^2
\end{align}
Because $Ļ$ 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 ...
2
votes
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 ...
0
votes
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 ...
-1
votes
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 ...
0
votes
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 ...
0
votes
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}$ ...
1
vote
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 ...
0
votes
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 = Ī±y$ and the second one is $y^2 = 2Ī±x$
What I tried is to solve the equation when the two curves are ...
0
votes
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 ...
1
vote
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:
-1
votes
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 ...
0
votes
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: 2ĻRh
However, a simple 8 radius circle is ...
0
votes
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 ...
4
votes
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 $\...
0
votes
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 ...
0
votes
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 ...
0
votes
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 ...
0
votes
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
-1
votes
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=\...
1
vote
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 ...
0
votes
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 ...
1
vote
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 ...
1
vote
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 ...
1
vote
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{...
0
votes
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= ...
2
votes
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 ...
0
votes
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....
0
votes
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}}}^{\...
3
votes
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 ...
0
votes
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 ...
0
votes
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 $A=\{(x,y) \in \mathbf{R}^Ā² \mid 3x^2+4y^2\leqslant12 \}$
I tried to use polar coordinates here, but couldn't get it to work.
If I have $x=r\cos(\theta)$ and $y=r\...
