Questions tagged [area]
Area is a quantity that expresses the measurement of the extent of a two-dimensional shape.
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Find the Area Bounded by $y = \frac{2}{π} [|\cos^{-1}(\sin x)| - |\sin^{-1}(\cos x)|]$ and $x$-axis between $ \frac{3π}{2}≤x≤2π $.
Find the area bounded by $\displaystyle y = \dfrac{2}{π} \left[ \space \left|\cos^{-1}(\sin x)\right| - \left|\sin^{-1}(\cos x)\right| \space \right]$ and $x$-axis between $ \dfrac{3\pi}{2}≤x≤2\pi $.
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How do I calculate the area of a triangle intersecting a cylinder?
I have a cylinder and a triangle in 3D space. Both can be arbitrarily positioned, oriented, and sized. I would like to know the area of the triangle that is intersecting the volume described by the ...
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Intuitive explanation of why the area under the curve of a hyperbola (1/x) is infinite but not the area of a decreasing exponential?
In some of the videos I've watched on the Laplace transform, the authors say that if the exponential is decreasing, the area calculated by the transform is finite, and in control theory we can say ...
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Prove that formula equals to the triangle's area
We need to prove that right triangles area equals to $AK \times KB$, first i tried to draw lines where sides touch circle and got formula $S = r^2 + AK\times r + KB\times r$, but it leads to nowhere. ...
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Find A, if A = Sum of areas bounded by y = $ e^{-x^3} $, $x=0,y=0,x=1$ and $y = e^{-x^2}$, $x=0,y=0,x=1$ [closed]
The only information about the answer i have is that [A] = 0 when [.] Represents the greatest Integer function.
My approach was using integration by parts.
Let $I_1$ = $ \int_{0}^{1}e^{−x^{3}}×1 dx $.
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Find $A$, if $A=a+b$ where $a/b$ is the ratio of area of region bounded by the curve $y=\tan x$ and curve $\frac{4x^2}{\pi^2}+y^2=1$ ($a,b$ are prime)
Find $A$, if $\,A=a+b\,$ where $\,a/b\,$ is the ratio of area of region bounded by the curve $\,y=\tan x\,$ and curve $\dfrac{4x^2}{\pi^2}+y^2=1\;$ ($a,b$ are prime).
My approach was that since the ...
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Exact area of $x<\sqrt{(x-\frac{1}{4})^2+y^2}-2(x-\frac{1}{4})^2-2y^2+\frac{1}{4}$ without using double integrals
In another post I asked for the exact area of $x<\sqrt{(x-\frac{1}{4})^2+y^2}-2(x-\frac{1}{4})^2-2y^2+\frac{1}{4}$
someone found a way to do it with double integrals, now I'm asking for proof ...
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Distance Halfway Around Track Between Inner and Outer Edges
Here is a problem from an old Algebra book that has me a little stumped. I am not clear exactly what they are asking for or how to solve it.
(Problem 60.) Show that T (the area of the Track) is the ...
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Exact area of $x<\sqrt{(x-\frac{1}{4})^2+y^2}-2(x-\frac{1}{4})^2-2y^2+\frac{1}{4}$
$x<\sqrt{(x-\frac{1}{4})^2+y^2}-2(x-\frac{1}{4})^2-2y^2+\frac{1}{4}$
I was able to get to the integral part (I'll see if you can get there yourself, if not I'll share how) but I cannot seem to ...
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Given graph of a function $f(x)$, find area under $f(x^2-1)$ from $-1$ to $1$
Source: MAT $2011$
Above is the graph of the function $f(x)$. It is required to find
$$\int_{-1}^{1}f(x^2-1)dx$$
So the function $f(x)$ can be split into two parts -
$f(x) = \left(x+1\right)$ for $-...
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How many unique triangles of the same type can fit in a circle?
Given a circle of any radius, i want to find how many unique triangles of the same shape can fit in that circle.
All the three vertices of every triangle have to be on the perimeter of that circle.
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How many solutions does $\int_{0}^{x}{\sin (\sin t)dt} = 0$ have?
Source: MAT $2012$
How many solutions for $0< x \le 2\pi$ does the following integral have?
$$\int_{0}^{x}{\sin (\sin t)dt} = 0$$
How would this curve look like, and how do I go about visualising ...
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Express the unshaded area as a fraction of total area [closed]
$1/7$ of a star overlaps with $2/9$ of a square and it's shaded. Express the area of the unshaded parts as a fraction of the total area of the figure.
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What is the equation and area under curve for Covid load dynamics?
Covid virions on infection, replicate exponentially and once the body's defense system starts attacking it then it also seems to decrease exponentially.
Source
The time period when the PCR test is ...
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Which is the area of intersection, using polar coordinate
For the curves $r=3cos\theta$ and $r=1+\cos\theta$ i need to find the area on the intersection of two curves. Now, $3cos\theta=1+\cos\theta$ implies that $cos\theta=\frac{1}{2}$ which said that $\...
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Calculate the area between $f(x) = x \cos^2(x)$ and $g(x) = x$
I want to calculate the area bounded by the curves $f(x) = x \cos^2(x)$ and $g(x) = x$.
To find the points of intersection, I set $f(x)= g(x)$.
$$\begin{aligned}x\cos^2(x)&= x\\
x\cos^2(x)-x&= ...
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Distance Between Objects Evenly Distributed In a Shape
The Milky Way's radius is 50,000 light years, 1,000 light years thickness. There are 865 stars, what is the distance between each of these stars?
I've searched for a while and nothing helped me. I'm ...
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Find surface area of cone cut off by a cylinder
Find the surface area of cone $ {x^2 + y^2 = z^2} $ cut off by surface of cylinder $ {x^2 + y^2 = a^2} $ above the $xy$ plane.
My approach:
I considered projection of the area on $xy$ plane cut off by ...
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Expected area of triangle by choose 3 points inside unit circle randomly (Disk Triangle Picking)
I tried to find the value of "Expected area of triangle by choose 3 points inside unit circle randomly". First time, I thought it is $\displaystyle\frac{2}{3\pi}=0.212$. But computer ...
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Find area of triangle given one angle and the lengths its altitude divides the opposite side into
In triangle $\triangle ADC$, $DB$ is perpendicular to $AC$ at $B$ so that $AB=2$ and $BC=3$ as shown in the figure. Furthermore, $\angle ADC=45^\circ$ . Find the area of $\Delta ADC$.
Tried the ...
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Lidded Box Optimization: Solve Unfolded Area Minimization Problem given its Dual Volume Maximization Problem
The above volume maximization problem is simple to solve. What would its dual, the unfolded area minimization problem look like, given fixed volume of $V=\frac{8000}{27} \approx 296.3$? I would hope, ...
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What is the area bounded by curve y²=4x,y axis and the line y=3.Solve using double integration. [closed]
What is the area bounded by curve $y²=4x$, $y$ axis and the line $y=3$. Solve using double integration.
I operated using double integration and I am getting $9/2$ sq units but the answer to this is $...
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How does one calculate the area of a set?
The set is $M=\{(x,y)\in\mathbb{R}^2:|x|+|y|\leq 1\}$.
Question: How do you calculate the area of $M$? More specific, how do you find the bounds of integration?
Attempt: I tried to solve the ...
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Area of a region defined by boundaries
Question: Find the area of the region whose boundary is defined by $\:y=\frac{1}{x},\:y=\:x^{\frac{2}{3}}, X-\text{axis},\:x=3$.
I am confused about which region to take for finding the area. Below ...
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Area bounded by a curve
Question: Find the area bounded by the curve $y$= $4$ $Tan{\frac{x}{2}}$, X-axis (or $y=0$), $x= {\frac{-π}{2}}$ and $x= {\frac{-π}{3}}$.
I am having a slight difference in the answer and I am not ...
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Effects of horizontal scaling on area under curve?
If f(x) is a continuous function and we stretch it horizontally by a factor n (multiply x by 1/n in the formula) then my intuition and some examples i solved tell me that if we look at a segment of ...
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Finding Area Between Two Graphs
I just finished the chapter in my textbook about finding the area trapped between two graphs. However, I have a few additional questions. To find the area, you usually find the definite integral of ...
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Need help in correctly framing problem to calculate the area $D$
Let $D: = \{(x, y) : |x| + |y| \leq 1 \}$
How to calculate area $D$?
I've calculated it as follows and it is wrong:
I've framed the problem as $-1 \leq x \leq 1$ and $-1 \leq y \leq 1$
$\int_{-1}^{1} \...
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The inverse problems of the pedal triangle.
Given a point $P$ inside the $\triangle ABC$, the pedal triangle of $P$ is the triangle whose polygon vertices are the feet of the perpendiculars from $P$ to the side lines.
As we know that if $P$ is ...
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How to calculate the intersection of the 2 different circle's area [closed]
I have two different circles with their formulas, which intersect from different parts of themselves, how can I calculate the area of the intersection and the rest of the circles? Is there some way to ...
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Area of intersection between hypercube and hyperplane
To be short, I have an hypercube $C_\delta = [0, \delta]^{n+1}$ and an hyperplane $H_t = \{x \in \mathbb{R}^{n+1} : \|x\|_1 = t\}$. The quantity I'm looking for is the hyper-area of $C_\delta \cap H_1$...
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Area of $x^{10}+y^{10}\leq 1$
Whilst looking at someone's vector calculus problem, they mentioned that, making use of Green's Theorem, they had to express the line integral of the boundary of $x^{10}+y^{10}\leq 1$ in terms of its ...
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Area Bounded Between $y^2=8x + 16$, $y^2=-4x + 28$ and the $x$-axis
I am just getting started with applications of integration, and I am stuck upon a footling question. Required to find the area bounded between $y^2=8x + 16$, $y^2=-4x + 28$, and the $x$-axis. The way ...
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Similar isosceles triangles
$ABC$ and $ADB$ are similar isosceles triangles, such that $BC: CD = x : y$
I am told that the area of $ABC$ : area of $ADB$ = $x$ : $x + y$, but I cannot see intuitively why this would be the case?
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Rectangle inside an ellipse that is inside a rectangle
Suppose there is a rectangle $ABCD$ with $A=(-2,1), B=(-2,-1), C=(2,-1), D=(2,1).$ Then we have an ellipse $\frac{x^2}{4} + y^2=1$ that is tangent to the rectangle. Then we build the rectangle $EFGH$ ...
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Why is the minimal surface of revolution not a cylinder?
I'm trying to find the curve $r=f(z)$ that links two parallel circles of same radii, aligned on the same axis $ (Oz) $, and that minimizes the surface of revolution between both circles.
The solution ...
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Finding area of quartic [duplicate]
Suppose we have a shape $x^4 + y^4 = R^4$. How can we find the area of this?
My initial thought was to find $\int_0^R\sqrt[4]{R^4 - X^4}dx$ and multiply this by 4, but I am struggling to find the ...
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Conservation of the area under a curve?
Imagine the function $\Psi(x, t) \in \mathcal C^2(U)$ with $U \subset \mathbb R^2$, obeys the following PDE:
$$
A\frac{\partial^2 \Psi}{\partial t^2} +B\frac{\partial^2 \Psi}{\partial x^2} + C\frac{\...
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Area of a triangle on a torus
On a sphere the area of a trinagle is $A=\dfrac{\pi r^2}{180^{\circ}}\cdot(\alpha+\beta+\gamma-180^{\circ})$ where $\alpha, \beta, \gamma$ are the angles of the triangle.
Is there a similar easy ...
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Can we find other triangles with different integer sides which have equal integer area and perimeter using this method?
A few days ago, I was reading this Wikipedia page and this part caught my eyes:
As W. A. Whitworth and D. Biddle proved in 1904, there are exactly three solutions, beyond the right triangles already ...
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Solving a kinematics problem involving areas under curve
The Original question is as follows:
Two cars $A$ and $B$ simultaneously start a race. Velocity $v$ of the car $A$ varies with time $t$ according to graph shown. It acquires a velocity of $50 m/s$ ...
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A cow is tied to the outside of a raised square platform of side 10m, with a rope of 25m. what is the area the cow can graze? [closed]
This is similar to a number of such quesiton, but the overlapping areas are a bit tricky. Ive a attached a diagram which i think is correct
(may not be)
overlapping bits
The file is also available ...
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MIT opencourseware 18.01 Single variable calculus. Integration(Areas and Volumes). Recitation problem
The question:
Find the volume of the solid generated by rotating the region bounded by y=0 and y=sqrt(x)
around the line x=6.
My approach (1):
integrate w.r.t to y.
integrate limit from 0 to sqrt(6) ...
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Calculating the average are of an annulus and the unit circle
I'm trying a problem and came across the following problem. Choose a point at random uniformly from $\overline B_1(0)$, let's say $x$. Fix some $t_1,t_2\in\mathbb R^+$, what's the average area of the ...
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How to calculate integral of $f(x)=\sqrt{x^3+2}$?
I just had a math test where I had to use trapezoid rule to estimate the area from $a$ to $b$ under the curve:
$$ f(x) = \sqrt{x^3+2} $$
Is there a way to find the exact area under the curve with an ...
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Determinants / find area of a triangle using the determinant method. In the picture below why is the area equal to 0
Why is the area of triangle equal to 0
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Point on parabola satisfying a condition
On parabola $p:y^2=4x$ find a point at which tangent and its normal at that point create a triangle of area $20$ with $x$-axis. Find equations of that tangent and its normal.
(Normal is a ...
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Attempting to compute *surface* of solid of revolution
I saw that in order to compute the volume of a surface of revolution, we can use $\int_a^b\pi f^2\left(x\right)dx$, where $f$ is the curve to be rotated. This seemed really intuitive: for each "...
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How do you find the area of the intersection of circles?
enter image description here
Thank you for your help.
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Area under two curves and between two curves are equal.
In the accompanying figure, y=f(x) is the graph of a one to one continuous function f. At each point P on the graph of y=2x^2, assume that the areas OAP and OBP are equal. Here PA, PB are the ...
