1
vote
2answers
28 views

Evaluating a polar double integral on the semi disc

The integral: $$\iint_D (x^2-y^2)\,dx\,dy$$ where $D$ is defined as: $$\{(x,y)\in \mathbb R^2 \mid x^2+y^2\le 1, x\ge 0\}$$ Context I have solved double integrals on quarter discs but this semi ...
-1
votes
1answer
35 views

Find finite area between curves [closed]

Find the finite area enclosed between $r= a \sin 4(\theta)$ and $r= a \sin 2(\theta)$ in polar coordinate system.
1
vote
1answer
62 views

Finding a mistake in the computation of a double integral in polar coordinates

I have to find $P\left(4\left(x-45\right)^2+100\left(y-20\right)^2\leq 2 \right) $ $f(x)$ and $f(y)$ are given, which I will use in my solution below . ...
3
votes
3answers
288 views

Evaluation of the integral of $e^{-(x^2+y^2)}$ over a disk

Show that $$\renewcommand{\intd}{\,\mathrm{d}} \iint_{D(R)} e^{-(x^2+y^2)} \intd x \intd y = \pi \left(1 - e^{-R^2}\right)$$ where $D(R)$ is the disc of radius $R$ with center $(0,0).$ I ...
3
votes
1answer
47 views

Use a double integral in polar coordinates to find the area

So the area is just an intersection of two circles Converting the two circles to polar coordinates, I get: $r(r-2\sin\theta)=0$, and $r(r-2\cos\theta)=0$ Ummm so $r =0$ and r = $2\sin\theta$ ...
0
votes
1answer
25 views

Compute double integral on polar coordinates, find $r(\phi)$

I have the function $f(x,y)=y^2-2x^2y+6x^3-3xy+2y-6x$ and the region $\{y\geq 2x^2-2, y\leq 3x\}$. The region is: To compute the integral in cartesian coordinates: ...
2
votes
2answers
64 views

Integrating $ \int_0^2 \int_0^ \sqrt{1-(x-1)^2} \frac{x+y}{x^2+y^2} dy\,dx$ in polar coordinates

I'm having a problem integrating $ \displaystyle\int_0^2 \int_0^ \sqrt{1-(x-1)^2} \frac{x+y}{x^2+y^2} \,dy\,dx$. I drew the graph, and it looks like half a circle on top of the $x$ axis. I tried ...
0
votes
3answers
56 views

Integrating $\int_1^2 \int_0^ \sqrt{2x-x^2} \frac{1}{((x^2+y^2)^2} dydx $ in polar coordinates

I'm having a problem converting $\int_1^2 \int_0^ \sqrt{2x-x^2} \frac{1}{(x^2+y^2)^2} dy dx $ to polar coordinates. I drew the graph using my calculator, which looked like half a circle on the x ...
0
votes
1answer
29 views

Tranforming to polar co-ordinates

$$I = \int_0^1\int_0^{\sqrt{1-x^2}} xy \, dy\, dx$$ By transforming to circular polar co-ordinates, evaluate I. How do I do this? Is there a formula/strategy for doing this that works with ...
0
votes
1answer
16 views

Center of Mass double Integral using polar Coord.

Find center of mass given Lamina pictured: https://s3.amazonaws.com/wamapdata/qimages/qtrring.gif with inner radius of 3 and an outer radius of 7, and a density function $$\rho(x,y) = ...
1
vote
2answers
29 views

Surface: intersection of 2 polar curves

I have these two polar curves: $$ C_1: r = 2 - \cos(\theta)\\ C_2: r = 3 \cos(\theta) $$ Plots: C1 and C2. I need to find the surface of $D = C_1 \cap C_2$. I started by finding the solution to ...
0
votes
2answers
40 views

how do we interpret this integral from polar co-ordinates

$$\text{Find } \int_C rdr$$ Where $C$ is any closed loop. I feel that the answer is zero, i have no hard reasoning. Here $r$ is the parameter from the polar coordinates.
1
vote
0answers
28 views

Double integral polar coordinate using substitution

I have to calculate: $$\iint _{R} \frac{x}{\sqrt{x^{2}+y^{2}}}dA$$ and R is this region: $$x^{2}+y^{2}=16; x^{2}+y^{2}=4; y = \sqrt{3}x; y=\frac{x}{\sqrt{3}}$$ so I used substitution: $$x = r\cos ...
4
votes
1answer
150 views

Multiple integral over a disc

I would need some help on this integration problem: $$I=\int_0^{2\pi}\int_0^{R}\int_0^{2\pi}\int_0^{R}\exp(-a\ r_{12}) \ r_1 \ r_2 ...
0
votes
1answer
35 views

Ellipse region in polar coordinates

if I want to write the region in $R^2$ bounded by the ellipse $$10x^2 + 17 y^2 = 29$$ In polar coordinates($x=r\cos \theta, y= r \sin \theta$), how can I find the limit of $r$?
3
votes
1answer
64 views

approximate this fancy looking double integral

$$\int_{0}^{2\pi} \int_{0}^{1}r^5\sin^22\theta\left(1-r^2 \right)^2\sqrt{1+\left(1+ \cos^2\theta \right)36r^2 }\hspace{1mm}drd\theta$$ I tried integrating myself, spent many hours but could not ...
0
votes
1answer
43 views

Volume of solid bounded by $z^2 = x^2 + y^2$ and $x^2 + y^2 = 2x$

Calculate the volume of the solid bounded by $z^2 = x^2 + y^2 $ and $x^2 + y^2 = 2x$ My attempt: Using cylindrical coordinates, $$ \mathrm{Vol} = \int_{-\pi/2}^{\pi/2} \int_0^1 \int_{-r}^{r} r ...
1
vote
1answer
38 views

Double Integral Help $(x^2+y^2+a^2)^{-2} dx \, dy$

Hi I'm currently revising for a maths module that I am taking as part of my physics degree. All was going well until I hit a dead-end with this integral, any ideas how to evaluate it? $$ ...
3
votes
1answer
46 views

Polar coordinates: $ \iint_D (\sqrt{a^2 - x^2 -y^2} - \sqrt{x^2 + y^2})\:\mathrm{d}x\:\mathrm{d}y$

I need to calculate the following integral $$\iint_D \left(\sqrt{a^2 - x^2 -y^2} - \sqrt{x^2 + y^2}\right)\:\mathrm{d}x\:\mathrm{d}y$$ where $D$ is the disk $x^2 + y^2 \leq a^2$ Using the ...
0
votes
2answers
77 views

Evaluate an integral using polar $\displaystyle\int_0^2 \int_{-4\sqrt{4-x^2}}^{4\sqrt{4-x^2}}(x^2-y^2)\,dy\,dx$

How do you evaluate the following integral using polar cordinates. $$\int_0^2 \int_{-4\sqrt{4-x^2}}^{4\sqrt{4-x^2}}(x^2-y^2)\:\mathrm{d}y\:\mathrm{d}x$$ I converted it to polar coordinate making it ...
4
votes
2answers
40 views

Finding Multivariable Limits

Is there any good way to find a multivariable limit other than switching to polar coordinates? For example, students each year are inundated with problems like $$\lim_{(x,y)\to ...
1
vote
1answer
75 views

Polar equation — find area under graph using double integral

What is the area of the region in the plane bounded by the curve given in polar coordinates $r = 4 + 2\cos(2\theta)$? Could someone walk me through the conversion of polar coordinates to rectangular ...
0
votes
2answers
35 views

Use Polar Coordinates to Find the Limit…

Use polar coordinates to find the limit. [If $(r, \theta)$ are polar coordinates of the point $(x, y)$ with $r \geq 0$, $r \to 0^+$ as $(x,y) \to (0,0)$)] $$\lim \limits_{(x,y) \to (0,0)} ...
1
vote
1answer
51 views

For which $\alpha \in \mathbb{R}$ does $\int_{\mathbb{R}^n} \big(1+|x|\big)^{\!-\alpha} \mathrm{d}x$ exist?

I assume only $\alpha \gt 1$ gives $\int_{\mathbb{R}^n} (1+|x|)^{-\alpha} \mathrm{d}x \lt \infty$ (simply because this is true for $n=1$). I also assume some clever transformation could be used for ...
3
votes
1answer
86 views

Change to polar coordinates when evaluating limits of functions in two variables?

I have a function in two variables $f(x, y)$ and need to calculate the limit $$ \lim_{(x, y) \rightarrow (2, 3)}{f(x, y)} .$$ If I decide to change to polar coordinates, how can I determine where $r$ ...
0
votes
1answer
44 views

Velocity of a particle in polar coordinates

The equations $r = 3\sin(2\theta)$ and $\frac{d\theta}{dt} = 2$ describe the motion of a particle in polar coordinates. Find the velocity of the particle in terms of the unit vectors $u_r$ and ...
1
vote
2answers
133 views

how to find existence and value of limit in multivariable calculus

I was in maths class and i found a question interesting. Find the limit of $\lim_{(x,y)\to (0,0)} \frac{2x}{x^2+x+y^2}$ if it exist.one of my friend did this question by transforming into polar ...
2
votes
1answer
57 views

Area of a sphere bounded by a paraboloid

I need to find the area of the surface $x^2+y^2+z^2 = a^2$ for $y^2 \ge a(a+x)$. I know that $A = 4a \int_{-a}^0 dx \int_{\sqrt{a^2+ax}}^{\sqrt{a^2-x^2}} \frac{dy}{\sqrt{a^2-x^2-y^2}}$, but I have ...
2
votes
1answer
86 views

Help with Polar coordinates and the length of the curve.

I have a test coming up today and I was going over our past midterms and this question came up. I tried it but its not working, please any hints or solution in how to do it will be really helpful. ...
2
votes
1answer
33 views

Polar Integral Confusion

Yet again, a friend of mine asked for help with a polar integral, we both got the same answer, the book again gave a different answer. Question Use a polar integral to find the area inside the ...
4
votes
2answers
63 views

Given integral $\iint_D (e^{x^2 + y^2}) \,dx \,dy$ in the domain $D = \{(x, y) : x^2 + y^2 \le 2, 0 \le y \le x\}.$ Move to polar coordinates.

Given integral $\iint_D (e^{x^2 + y^2}) \,dx \,dy$ in the domain $D = \{(x, y) : x^2 + y^2 \le 2, 0 \le y \le x\}.$ Move to polar coordinates. First of all I tried to find the domain of $x$ and ...
2
votes
2answers
649 views

Volume of a sphere (r=2a) with hole(r=a) drilled through centre, using spherical polar coordinates.

Need help solving 11.bi), A cylindrical hole of radius a is bored through the center of a sphere of radius 2a. Find the volume of the remaining material, using spherical polar coordinates. (You ...
0
votes
3answers
198 views

Changing from Cartesian coordinates to Polar coordinates

Rewrite the iterated integral $$\int_0^1 \int_0^{\sqrt{2y - y^2}} (1 - x^2 - y^2)\,dx\,dy$$ in polar coordinate form. Do not evaluate the integral. Here is my answer: ...
2
votes
7answers
2k views

Why does $r=cos\theta$ produce a circle?

I am trying to do a double integral over the following region in polar coordinates: I know that the limits of integration are: $$\theta=-\pi/2\quad to\quad \theta=\pi/2\\r=0\quad to\quad ...
2
votes
1answer
184 views

Use polar coordinates to find the volume of the given solid

Use polar coordinates to find the volume of the given solid bounded by the paraboloid $z=1+2x^2+2y^2$ and the plane $z=7$ in the first octant. I did it. Is that right ? $$\int_0^{\pi \over 2} ...
1
vote
0answers
37 views

Polar coordinates: fixing ratio between arc and radius

When drawing a coordinate system with fixed step size, the standard polar coordinates $$ x=r\cos(\theta), y=r\sin(\theta) $$ exhibits stretched pixels for large $r$. Ignoring the singularity in ...
1
vote
0answers
162 views

How to plot a stream function

This question relates to fluid mechanics and I have the components in polar coordinates. The components of the velocity field are; $$v_r= \frac{-kr}{z}$$ $$v_z= kz$$ $$v_\theta= 0$$ and I have ...
0
votes
0answers
158 views

Laplace in Polar Coordinates

I'm working on the Laplacian in Polar conversion form Cartesian and I'm having trouble understanding an operation found in the description at ...
2
votes
0answers
93 views

Polar Coordinates for Multivariate Limits With Three Variables

When working on limits with two variables, $f(x,y)$, I like to convert the problem to polar coordinates a lot of the time, by changing the question from $$\displaystyle\lim_{(x,y)\to (0,0)}f(x,y)$$ to ...
0
votes
1answer
49 views

change of variables while integrating

Suppose I have an integral that looks like: $$I=\int_{r=0}^\infty\int_{\omega_1=-\infty}^\infty\int_{\omega_2=-\infty}^\infty ...
1
vote
2answers
41 views

Find the image of a ring

I'm working on the following problem: Find the image of the ring defined by $4 \lt x^2 + y^2 \lt 16 $ under the mapping $$F(x,y) = \left(\frac{x}{x^2+y^2} , \frac{y}{x^2+y^2}\right)$$ It looks to ...
2
votes
1answer
198 views

How do I define the limits of a double integral in polar coordinates over an annulus?

Evaluate the double integral by re-writing them in polar coordinates: $\displaystyle\iint\limits_{R}\frac{y^2}{x^2}\ dA$, where $R$ is part of the annulus (ring) $9\leq x^2+y^2\leq 25$ lying ...
2
votes
1answer
188 views

How to integrate over polar coordinates

Evaluate the following double integral by rewriting it in polar coordinates: $\displaystyle\iint\limits_Dxy\,dA$, where $D$ is the disc with center at the origin and radius 5 I have very little ...
1
vote
0answers
187 views

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, $\phi(r)$ ...
2
votes
1answer
60 views

Area of $\left( \frac{x^2}{9}+\frac{y^2}{25} \right)^2 \le x^2 + y^2$

I've used the modified polar coordinates: $x = 3r \cos \theta$, $y =5r \sin \theta$, which got me to $$r^2 \le 9 \cos^2 \theta + 25 \sin^2 \theta$$ What now?
3
votes
3answers
1k 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 ...
1
vote
1answer
500 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 ...
1
vote
2answers
111 views

$f(x,y)=\langle y- \cos y, x \sin y\rangle$

$f(x,y)=\langle y-\cos y,x\sin y\rangle$ $C$ 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). ...
1
vote
1answer
178 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
2
votes
1answer
114 views

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 ...