# Tagged Questions

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### Polar curve $r = 2\cos \theta -1$

$$r = 2\cos \theta -1$$ I am suppose to find the polar curve of the inner loop. Here is its graph, courtesy of Wolfram|Alpha, I am having trouble working out this polar function on a cartesian ...
1answer
78 views

### Need a hint on what's wrong - polar coordinates

I'm asked to solve the following $$\int^2_0 \int^\sqrt{4-y²}_0 \sqrt{4-x^2-y^2} dxdy$$ I thought about using polar coordinates: (1) $0 \le x \le \sqrt{4-y^2}$ is the upper half of a circumference ...
2answers
220 views

### How to verify a conversion to spherical coordinates?

Is it possible to verify if a conversion of an integral in Cartesian coordinates to spherical coordinates was done correctly other than revising it looking for mistakes? I mean, is there some kind of ...
0answers
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### Polar Integration over intersection of two circles

Let $C_0$ denote a circle centered at $(0,0)$ with a radius of $r_0$ and let $C_1$ denote a circle of radius $r_1$ centered at a point $(x_1,0)$. Assume that we are given some function, $\phi(r)$ ...
2answers
142 views

### limits of Surface area of revolution in polar co-ordinates.

My Question is Find the area of the surface generated by revolving the right-hand loop of the lemniscate $\;r^2=\cos2\theta\;$ about the vertical line through the origin (y-axis). I know the formula ...
3answers
591 views

### Why is the formula for the area of a cardioid $\int_a^b \frac{1}{2} r^2 d \theta$

I've seen this expression in many places :$\int_a^b \frac{1}{2} r^2 d \theta$ and was wondering if someone can explain where this came from? I've noticed that it's sometimes explained in conjunction ...
1answer
396 views

### Find the area of the shaded region between $r=e^{\theta/2}$ and $r=θ$ .

That's the picture of the shaded region I have to find the area of. I'm totally stuck on this problem mainly because these two curves don't intersect so I'm not sure how to find the bounds of ...
2answers
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### $f(x,y)=\langle y- \cos y, x \sin y\rangle$

$f(x,y)=\langle y-\cos y,x\sin y\rangle$ $C$ is the circle $(x-3)^2 + (y+4)^2 = 4$ orientated clockwise. Relevant theorems: Green's theorem (this is under the Green's theorem section of our book). ...
2answers
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### Integration of radial functions?

Let $f(|x|)$ be a integrable radial function in $\mathbb{R}^n$ ($|\cdot|$ denotes the euclidean norm as in convention). The following identity is used to simplify computations ...
1answer
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### Finding area between two polar curves using double integrals

I have a homework question that is asking me to find the area that lies: Inside the curve $r=2+cos(2\theta)$ But outside the curve $r=2+sin(\theta)$ I think I'm supposed to be using a double ...
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154 views

### Moment of inertia of a circle

A wire has the shape of the circle $x^2+y^2=a^2$. Determine the moment of inertia about a diameter if the density at $(x,y)$ is $|x|+|y|$ Thank you
1answer
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### Line integral of $F = r \times k$ on hemisphere

Exam revision - Verify Stokes theorem directly by explicit calculation of the surface and line integrals for the hemisphere $r=c$, with $z \geq 0$, where $F = r \times k$ and $k$ is the unit vector ...
1answer
156 views

### Polar Coordinates: Dividing by the variable “r.”

Evaluate the iterated integral by converting to polar coordinates: $\large \int^2_0 \int^{\sqrt{2x-x^2}}_0 xy~dy~dx$ I successfully completed most of the problem; however, I had difficulty ...
1answer
662 views

### How to calculate the area between 2 polar curves: $r=\frac{4}{2}-\sin\theta$ and $r=3\sin\theta$?

How to calculate the area between 2 polar curves: $r=2-\sin\theta$ and $r=3\sin\theta$? I know that one curve is a limaçon and the other is a circle. I have them drawn out as well, my only question ...
2answers
78 views

### Integration, polar coordinates

My question is general rather than specific.If a problem requires to find the area of a figure bounded by a curve given in polar coordinates,how do we find the limits of integration analytically ...
1answer
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### evaluation of double order integral using polar co-ordinates

When evaluating double integral using polar co-ordinates, does the order of $dr ~ d\theta$ make any difference? Suppose, $$\int_0^{\pi/4}\int_0^{\sin\theta} r^2 dr d\theta$$ ...
1answer
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### How to calculate a double integral over a triangle by transforming to polair coordinates & by using a transformation

Let T be the triangel with vetrices $( 0,0 ) , ( 1,0 )\mbox{ and } ( 0,1 )$. Evaluate the integral : $$\iint_D e^{\frac{y-x}{y+x}}$$ a) by transforming to polar coordinates b) by using the ...
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### Transform to polar the following integral $\int_0^6\int_0^y x \, dx \, dy$

I need to transform this integral $\int_0^6\int_0^y x \, dx \, dy$ to polar and then find its value. I'm stuck finding the r-limits of integration.
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### Changing Variables in double integral

I have these particular exercise that i cannot solve. I know i have to change the variables, but i cannot figure out if i should use polar coords or any other change. Let D be the region delimited ...
1answer
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### Stokes' and Divergence Theorem Problems

I have 2 questions on stokes and divergence theorem each. I think I have done both and I just want to make sure that I did them correctly. Question 1 Let $C$ be the boundary of the surface ...
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### Explain this limit of integration for radius in polar coordinates.

Use polar coordinates to find the volume of the given solid: Inside both the cylinder $x^2 + y^2 = 4$ and the ellipsoid $4x^2 + 4y^2 + z^2 = 64$ The limit of integration for the radius goes ...
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620 views

### Mass and center of mass of lamina in polar coordinates

I need some help with the following problem which is question number 15.5.4 in the seventh edition of Stewart Calculus. Here is the problem definition: "Find the mass and center of mass of the ...
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### Evaluate the integral by converting to polar coordinate

$$\int^{\pi/2}_{\pi/4} \int^{\sqrt{2-y^2}}_y 3(x-y) dx dy$$ I attempted the following: $$\int_{\pi/4}^{\pi/2} \int_{0}^{1} 3r^2 (\cos\theta - \sin\theta) dr d\theta$$ which is wrong apparently. ...
1answer
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### Find the area of polar coordinates

The questions is to find the area in the bounded region in polar coordinates $r = \sqrt{\theta}$ from $3\pi/2$ to $2\pi$ Here is what I did: I got the integral of $\cfrac{1}{2}\theta d\theta$ from ...