For questions about area of plane figures.

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How do I determine for given two rectangles A & B, with B having constant dimensions, that B is more than 50% inside A? [duplicate]

I have 2 rectangles in a 2D plane, only in the first quadrant with limit 2000 x 2000: A (black) & B (red) Look at the attached image Both can change positions (coordinates, X & Y). B has ...
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21 views

why this formula doesn't work?

L=integral from theta(start) to theta (end) r d(theta )......to find the arc length we usually use L= integral from theta(start) to theta (end) sqrt r^2+(dr/dtheta)^2 dtheta why the first formula ...
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1answer
33 views

How is the area of a set of points in $\Bbb R^2$ defined?

Let $S$ be a subset of $\Bbb R^2$. If no vertical slice of $S$ contains gaps, we could define the area of $S$ through the following. $$A(S) = \int_{-\infty}^\infty\left(\sup\{y\in\Bbb R\mid (x,y)\in ...
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28 views

Area swept out by non-solar focus not same over equal time?

Per Kepler's laws, the area swept out by a line between the sun and a planet is equal for a given period of time. The sun is also one focus of the planet's elliptical orbit. What about the area swept ...
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1answer
12 views

Dealing with negative areas— coordinate geometry

Question: Find the area of a quadrilateral in the Cartesian plane, whose vertices are (-4, 5), (0, 7), (5, -5) and (-4, -2) My solution: [I meant to draw ...
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1answer
58 views

Prove the ratio between the following areas is 2 [closed]

We have two non parallel lines. The distance between each points of each line is the following: $AB=CD=EF=HG=1$ and $BC=GF=2$. Prove that the area of $ADEH$ is twice the area of $BCFG$. Here is a ...
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31 views

Is there a formula for calculating the area of 2d shapes on a sphere?

Let's say I have 8 90° triangles on a sphere, like this, where all the angles are 90° when measured: I know that the area of one of those triangles will be (4πr2) * 1/8 as each triangle will take ...
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2answers
19 views

Question on volume of swimming pool

Swimming pool is of length 20m, wide 5m and height of the swimming pool increase from the 1.6m to 4.4m. What is the volume of swimming pool? How I approached: Area of swimming pool = Area of cube + ...
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3answers
51 views

surface area of the solid (column side)

I made a problem But I'm stuck in solving .. :-( the problem is following. Find the surface area of the solid that lies under the paraboloid $z =x^2 + y^2$, above the $xy$-plane, ...
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2answers
111 views

Finding the area of a implicit relation

Let's say we have the function: $$x^2+y^2+\sin(4x)+\sin(4y)=4$$ I haven't taken Calculus III, in fact I'm just taking Calculus I. Since I learned how to find the derivative of implicit relations I ...
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1answer
24 views

Solving for surface area of a cylinder.

I am doing an online course and I have to solve for the surface area of a cylinder, the Volume is give and I must find the most efficient cylinder, the cylinder with the least surface area, an ...
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2answers
82 views

What is the correct definition of Area?

How is the area of a rectangle: length $\times$ breadth? We know that other areas can be derived from it. Also, the area under curves uses the area of rectangles as a basis.
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1answer
25 views

Fraction of a square area within two lines

I have a general line $r: ax + by + c = 0$ and two parallel lines $s,t$ distant $d$ from $r$. And a square of side $l = 1$ centered at $(x_c,y_c)$. The square sides are perpendicular to the $x$ and ...
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2answers
44 views

How would I calculate the area of a rectangle on a sphere using vertical and horizontal angles?

Imagine a sphere being one's eyeball and the rectangular area being the picture of one's view. Like putting a name tag sticker on a balloon. How can I find the area of the rectangle on the sphere?
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1answer
29 views

area bounded by the curves $y=\left |x-1 \right |$ and $x^2 +y^2=2x$

Find the area bounded by the curves $y=\left |x-1 \right |$ and $x^2 +y^2=2x$.(Where, $y \geqslant0 $) I know the general rule of solving this problem.But the limit is so peculiar.Is there any ...
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1answer
22 views

Area under the curve uing 4 rectangles evaluated at the right hand endpoint

Approximate the area under the curve $f(x)=3x^2+1$ over the interval $[1,3]$ using 4 rectangles evaluated at the right hand endpoints. Would I do $(3-1)/4=.5$? If so what do I do after that step?
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Find the area of the portion of a plane inside the cylinder

How can I calculate this? I think at some point I will need to use symmetry and change this to polar coordinates. In that case my radius is $\pi$, and $\theta=2\pi$ to 0. I can calculate 2 ...
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26 views

Double integral over rect

I am having issues solving this. I have to take the double integral over: $$ \int_D \int_D dA,\, D = \{ (x, y) \mid {} \leq y \leq 1, 0 \leq x \leq y \}$$ There is only one D below both ...
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1answer
37 views

How to express using integrals the area of the region bounded by $xy=6, y=2, y=6, and x=6$ to find volume of region

Find the volume of the solid generated by revolving the region bounded by the above graphs of the equations around the line y=6. My Attempt $\int_1^6(6- 6/x)) ^2 dx$ Why would we need two integrals? ...
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12 views

Find value of area of semicircle cross-sections for the lines x+2y=8, the x-axis, and the y-axis

The base of a solid is a region in the $1st$ quadrant bounded by the x-axis, the y-axis, and the line $x+2y=8$. If the cross-sections of the solid perpendicular to the x-axis are semicircles, what is ...
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2answers
22 views

Find area of plane region bounded by 2 curves

curves: y=x+3, x=-y^2+3 My Attempt: $\int_0^{1.7320508} (-y^2+3)-(y-3) dy$ + $\int_{-1.732051}^{0} (-y^2+3)-(y-3) dy$ = 17.320 My teacher's answer is 125/6. What's my error?
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1answer
29 views

How to compute the integral of that area?

The area is given by $ 0 \le x+y \le 4-(x-y)^2 $ ? By magic the inequalities were transformed into $ 0 \le \sqrt 2 \ u \le 4 - 2v^2 $ and after that computing the integral became almost trivial. I ...
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1answer
20 views

troublesome area problem

Calculate the area of the region of the graph bounded by: $$\begin{eqnarray} y &=& x \\ y &=& x^2 + 1 \\ y &=& 2 \\ x &=& 0 \end{eqnarray}$$ My final result is ...
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1answer
10 views

Pre-Calculus Domain & Discontinuity

I'm doing some corrections on a test and this question has me stumped: David wants to create a rectangular holding area for his baby dragons, but only has 100 feet of fencing. He is going to use the ...
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1answer
37 views

Problem with shaded regions in a square

The sides of a square are 16 cm in length. The midpoints of the sides of this square are joined to form a new square and four triangles (diagram 1). The process is repeated twice as shown in the ...
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1answer
19 views

Volume of solid

Find the Volume of the solid whose cross sections perpendicular to the $x$-axis are squares one side of which stretches from the graph of $y = 2x + 1$ to $y = −x$ for $0 ≤ x ≤ 1$.
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1answer
18 views

Does changing side order of quad change area?

I have a quadrilateral with side lengths $10.40$, $12.33$, $11.75$, $11.50$. I am not given any other information, no angles or anything. I do not need to find the area, since I know it is ...
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1answer
27 views

what is the area of the polygon with given constraints?

What is the area of the polygon formed by all points $(x, y)$ in the plane satisfying the inequality $ ||x| – 2 | + | |y| – 2 | ≤ 4 $ ?
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1answer
15 views

Find the area of the triangle formed

Find the area of the triangle formed by the x-axis, the y-axis, and the tangent line to the graph of $y = \frac{1}{\sqrt{x}}$ at the point where $x=k$. I have no clue how to do this and my teacher ...
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1answer
26 views

Using Derivatives and Tangent Line to Find Area

Let $(a, b)$ be an arbitrary point on the graph of $y=\frac1x$ ($x>0$). Prove that the area of the triangle formed by the tangent line through $(a,b)$ and the coordinate axes is $2$ square units. ...
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1answer
30 views

Circle problem and area [closed]

What is the area of the circle centered at the origin with radius $5$, restricted to the domain where $x>0$ and $y>0$.
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1answer
43 views

Area between curves $y=x^3$ and $y=x$

I've tried to done one of my homework problems for several times, but the answer doesn't make sense to me. The question asks to find the area between $y=x^3$ and $y=x$. Those are odd functions, and ...
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1answer
14 views

Area between $y=2+|x-1|$ and $y=-\frac{1}{5}x+7$

Question 17, page 448, from Anton 8th. The question asks for the area, and the answer is 24. Now, I did draw the graph, found the points where both functions touch each other by $f(x) = g(x)$: ...
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2answers
24 views

Finding Area of region between cruves

I'm given $$y = \frac{4}{1+x^4} $$ and $$y = 2x^2$$ I know I have to integrate, but I'm not really sure how to find the limits to integrate at.
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2answers
41 views

Area between $y = \frac{2}{1+x^2}$ and $y = |x|$

The exercise is from Anton 8th, page 443, question 15. The problem asks for the area between: $f(x) = \frac{2}{1+x^2}$ and $g(x) = |x|$ It is not said what interval the area should be calculated, so ...
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2answers
46 views

Regular Octagon Area

Doing some maths homework I came across the area of a regular octagon on Google. This was given by: $$ A=2(1+\sqrt{2})a^2 $$ I thought this looked rather ugly and slightly complicated and so began to ...
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2answers
38 views

Area between $y=x^4$ and $y=x$

The problem I'm having some trouble solving is this: calculate the area between $y=x^4$ and $y=x$. The points are $a = 0$ and $b = 1$, but the definite integral is negative. What am I doing wrong ...
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1answer
58 views

Areas between intersecting chords

In the circle below let the two chords be called $C_1$ and $C_2$, and their intersection be some point that is not the center. The chord power theorem tell us that $a \cdot b = c \cdot d$. I am ...
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25 views

I still could not figure out

IT is our homework problem but I have already submit it. Today, I asked professor, but I still could follow what he said clearly. $\frac{dX}{dt} = \mu(x)$ and $X(0;x) = x$, where $x,X\in R^n$ For ...
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0answers
11 views

Create equal areas with the intersection of a function with the first and fourth quadrants.

Here's a question I wasn't able to figure out in class. The first part was easy enough, but the second part stumped even my teacher as well. "Consider the first- and fourth quadrant regions between ...
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1answer
25 views

Ratio of area between similar triangles

This question has nearly no information and I've been stuck on this for quite some time. I tried drawing the median from A thru G but the 1x to 2x ratio didn't seem to help.
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2answers
65 views

Volume, Lateral Area, and Surface Area of an Elliptic Conical Frustum

What are the formulae for the volume, surface area, and lateral area (i.e. the surface area without the bases) for the above illustrated elliptic conical frustum? I think I've got the volume figured ...
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1answer
30 views

Using differentials with volume of a cube

my question is The volume of a cube is increased from 1000 cubic centimeters to 1156 cubic centimeters. Use differentials to determine. the side length of the cube increases by? the surface area ...
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1answer
65 views

Geometry problem.

I have to find what is theta($\angle$GOE = $\angle$CDE). Here is a condition for above shape: The shape OCG is a quarter of unit circle(center is O). The line DF is a tangent line of curve CG ...
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0answers
35 views

Area of a circle of Radius “r” in a rectangle

This is a very basic problem but i would like to ask as i am unable to resolve it. I have a rectangle of the following dimensions. $Length = L$ $Width = W$ I picked a point ($x,y$) in this ...
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2answers
47 views

Help with writing an Equation for area?

You have 240 feet of wooden fencing to form two adjacent rectangular corrals. You want each corral to have an area of 1000 square feet. So far I have a drawing of a large rectangle, split by a line ...
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0answers
17 views

floor plan optimization?

I have sheets of mesh $7.05\text{ m} \times 2.62\text{ m}$. I have a floor area of $17.5\text{ m} \times 24.5\text{ m}$. What is the best way to calculate optimal laying/orientation of mesh? If ...
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2answers
30 views

How do I begin to prove the limit of this definite integral?

I was fooling around with a graphing calculator, and I noticed a pattern in the functions of $$x^a+y^a=1$$ where $a$ is an even number. As $a$ increases, the graph begins to look like a square (if you ...
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2answers
60 views

Find the limit of a Riemann Sum

The function is $f(x) = 1-x^2$. I'm stuck as I can't factor the expression in the last line to find the limit.
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25 views

How to factor $(1-(i^2/n)(1/n)$ to isolate $i^2$ and form a sigma identity?

given sigma from $i=1$ to $n$ of $(1-(i^2/n)^2)(1/n))$ how would you factor this function to isolate $i^2$ and get $[n(n+1)(2n+1)]/6$ ? update... I got until the limit as n approaches infinity (1/n) ...