# 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 ...
1 vote
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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 ...
57 views

### 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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### 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 ...
92 views

### 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 $50 m/s$ ...
64 views

### 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) ...
1 vote
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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 ...
1 vote
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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
1 vote
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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 "...