1
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1answer
17 views

Using Polar Integrals to find Volume of surface

Here's the Question and the work that I've done so far to solve it: Use polar coordinates to find the volume of the given solid. Enclosed by the hyperboloid $ −x^2 − y^2 + z^2 = 61$ and the plane $z ...
0
votes
2answers
45 views

$r=5 \sec(\theta)$ into rectangular

I need to convert $r=5\sec(\theta)$ into rectangular form. I think I need to multiply both sides by $r$, because $r^2 = x^2 + y^2$, but i'm not sure how to convert $r=5\sec(\theta)$ in terms of $x$ ...
0
votes
2answers
36 views

Complex number polar form equation

I've been struggling with a complex numbers algebra question for a few days now, and the tutor says I still haven't got it right. Express $z_4 =−\sqrt{3} + i$ in polar form. Hence solve the ...
0
votes
1answer
177 views

Function psi, vector potential, satisfying conditions

Using spherical polar coordinates ($r, \theta, \phi$) verify that the vector $F = r^{-2}e_r$ is solenoidal. Find the function $\psi(r, \theta)$ such that $A = \frac{\psi(r, \theta)}{rsin ...
2
votes
0answers
39 views

Wrong answer within 'Calculus Solution Manual, Michael Spivak, 3rd ed'

I have a problem with the answer provided in the solution manual of Calculus, Michael Spivak, 3rd ed, The Problem: Consider a hyperbola, where the difference of the distance between the two foci is ...
0
votes
1answer
77 views

How to calculate the polar arc length of the entire cardioid $r=a(1-\cos\theta)$

I'm having a bit of an issue calculating the arc length of $r = a(1-\cos\theta)$. I'll begin by listing the steps I made in my attempt to solve this exercise. We know that the arc length formula ...
0
votes
1answer
24 views

Polar to phasor

Let's say that there is a polar equation: -2400 + 8320j To convert this polar equation to phasor form, should the negative be considered when trying to find the angle? Would the angle be +73.91 or ...
0
votes
0answers
29 views

Integration over the unit disk in polar coordinates

Let $D$ be the unit disk $x^2 + y^2 \le 1$. Evaluate $\int \int_D e^{x^2 + y^2} dx \, dy$ by making a change of variables to polar coordinates. My solution: I substituted $x = r \cos{\theta}$ and $y ...
1
vote
1answer
38 views

Find rectangular equation of a cardioid

Given the equation in polar form $$r = 1 - \sin\theta,$$ find the rectangular equation. So far, I found: $$x^2 + y^2 = 1 - 2\sin\theta + \sin^2\theta\quad x = \cos\theta - \sin\theta\cos\theta\quad ...
1
vote
3answers
114 views

How to calculate $\theta$ when we know $\tan \theta$.

Hej I'm having difficulties calculating the angle given the tangent. Example: In a homework assignement I'm to express a complex variable $z = \sqrt{3} -i$ in polar form. I know how to solve this ...
0
votes
1answer
30 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 ...
0
votes
1answer
291 views

Area that lies inside both curves: $r=sin2\theta, r=cos2\theta$

My integral is setup as: $$A=8\int_0^\frac{\pi}8{\frac12sin^22\theta}\space d\theta - 8\int_{\frac{\pi}8}^0{\frac12cos^2 2\theta}\space d\theta$$ $$=8\int_0^\frac{\pi}8{\frac12sin^22\theta}\space ...
1
vote
0answers
41 views

Derivatives of polar coordinates

I've got a problem for which I'm trying to calculate $\ddot r$. The problem is right here for the sake of reference. So far, I've got that: $$\ddot r=\frac{d}{dt}v_r=\frac{dv_r}{dr}\frac{dr}{dt}$$ ...
2
votes
1answer
28 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 ...
0
votes
1answer
41 views

Can I solve for the fractional volume of a hyperboloid?

This looks like a homework problem because it is. I'm stuck at the portion where I solve for fractional volumes. Suppose you are a part of a team designing a water tank in the shape of a hyperboloid. ...
2
votes
1answer
53 views

Problems with integrals, polar coordinates.

I am having problems parameterizing these integrals: $$\int_A{\frac{x}{1+x^2+y^2}}\mathrm{d}x\mathrm{d}y$$ for $A = \mathbb R^2 \bigcap \,\{y \ge 0\}$ and the volume of $M = \{(x, y, z) \in \mathbb ...
1
vote
2answers
61 views

How to solve a complex equation $w^4 = \sqrt{3} -i$

$z = \sqrt{3} -i$ How do I solve a complex equation $w^4 = \sqrt{3} -i$ I know that I first have to rewrite z to polar format which I have done as $z = 2(cos (-π/6) + sin (-π/6))$ but I do not know ...
0
votes
1answer
27 views

Path of an ellipse

A path is described by the position vector $\mathbf{r}$: $$\mathbf{r}=a\cos(\omega t)\mathbf{\hat{i}}+b\sin{\omega t}\mathbf{\hat{j}}$$ I am asked to show that the path is the ellipse in the form of: ...
0
votes
1answer
554 views

Can someone check my answer for this area between 2 polar curves question?

Find the area of the region that lies inside the circle $r = 1$ and outside the cardioid $r = 1-cos(\theta)$ I drew the graph and set it up like this: $$ \int_0^\pi \frac{1}{2} [ (1)^2 - ...
1
vote
2answers
236 views

Evaluating the area in the polar coordinates

So the problem asked me to find the area of the region that lies inside both of the circles $$r=2sin\theta, \quad r=sin\theta +cos\theta $$ I know that $r=2sin\theta$ is $x^2+(y-1)^2=1,$but ...
1
vote
2answers
37 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
79 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 ...
2
votes
2answers
85 views

How to find the number of intersection for $ \rho =\frac{\theta} {2\pi+1} $ and $\rho =\frac {1} {2-\cos\theta} $

How to Find the number of intersection for curve $ \rho =\frac{\theta} {2\pi+1} $ and curve $\rho =\frac {1} {2-\cos\theta} $ .
0
votes
1answer
67 views

Another polar integral bounds question.

A plane region $R$ is determined by the inequalities $y\ge0$, $y\ge-x$, $x^2+y^2\le3\sqrt{x^2+y^2}-3x$. Sketch the region and find it's area. I have foregone sketching the area and tried to use ...
1
vote
2answers
93 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
130 views

How did theta become equal to 3pi/4 here?

How did theta become equal to 3π/4 in this particular example? Find a set of polar coordinates (r,θ) of the cartesian point (-4,4) such that -2π ≤ θ ≤ 2π and a. r > 0 and θ > 0 b. ...
1
vote
1answer
614 views

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 ...
2
votes
2answers
171 views

Find Cartesian equation of $r=\theta$

I solved this problem, but I'm not sure my answer is correct as it seems very complex (compared to the polar equation). Did I make some mistake along the way or is it the right solution? $$r=\theta$$ ...
2
votes
1answer
2k views

Find a Cartesian equation of $r = 4\cos\theta$

I was able to figure the substitutions inside the equation, but I'm stuck with the equation's manipulation that will give me the solution. What would be my next step? $$r = 4\cos\theta$$ $$r^2 = ...
4
votes
2answers
200 views

Parametrization of a curve in polar coordinates

I'm trying to change this parametrics equations to polar coordinates $$ X(t) = 2\cos(t) - \sin(2t) \\ Y(t) = 2\sin(t) - \cos(2t) $$ What i tryed to do was raise the two equations squared, sum ...
3
votes
1answer
74 views

polar coordinates ..question about the answer from the solution manual

Im trying to figure out but for some reason I dont know how to...could someone please tell me how did they get this answer from the solution manual....they skipped steps so I have no idea
1
vote
2answers
4k views

Find the area of the Rose's petal.

If a Rose leaf is described by the equation $r = \sin 3\theta$, find the area of one petal.
0
votes
3answers
135 views

Cartesian and Polar Coordinate

I should give the Cartesian Coordinates $(x,y)\in \mathbb{R\times R}$ and Polar Coordinates $(r,\varphi)\in R^+\times [0,2\pi)$ of the following Complex Numbers: a) $z_{1}=-i$ b) $z_{2}=\sqrt{3}+i$ ...
1
vote
2answers
59 views

Polar coordinates - issue with direction denoted by angle

Convert $1-\sqrt{3}i$ to polar coordinates $(r,\varphi)$. I started by computing $r=|1-\sqrt{3}i|=\sqrt{1^2+\sqrt{3}^2}=\sqrt{4}=2$. When I tried to compute the angle I did something like ...
1
vote
1answer
654 views

Cardioid given by the polar equation $r = 1 − \cos(\theta)$

Let $C$ be the cardioid given by the polar equation $r = 1 − \cos(\theta)$ , $−\pi \le \theta \le \pi$. (a) Find the intersection points of the curve with the line $\theta = \pi/4$. (b) Find the ...
1
vote
0answers
216 views

(Calculus 3) Having trouble finding the polar equation of a hyperbola.

Eccentricity e=sqrt(2), and one vertex is located at (2,0). I do know that if the vertex is located at (2,0), then the directrix is 2 units from the vertex. I am not sure how to find the location of ...
0
votes
1answer
4k views

Magnitude of average velocity..

A particle is at the position with Cartesian components $(5.0 m, 4.0 m)$ and $5.0 s$ later it is at the position with components $(8.0 m, 0.0 m)$. What is the magnitude of its average velocity? I ...
2
votes
1answer
204 views

The point C has coordinates (2,k) and the distance from A to C is 5. Find the 2 possible values of the constant k

The coordinates of A are (6, -2) I have managed to do every other question in this section but this one, my brother tried to help but just drew a graph and the answers without explaining it, i dont ...
1
vote
4answers
299 views

Find the length of a curve

Find the length of the following curve: $r(t) = e^{-t} \sin(t)+e^{-t}\sin(t) i$ for $0 \leq t\leq 1$. Any ideas?
1
vote
1answer
3k views

Find the equation in polar coordinate form for a straight line through the points with polar coordinates (4,0) and (4,π/3).

Find the equation in polar coordinate form for a straight line through the points with polar coordinates $(4,0)$ and $(4,π/3)$. Here's my steps: 1.Write the two points in cartesian coordinates: the ...
1
vote
1answer
174 views

Maximum point of a polar function

I have a curve C with polar equation $$r^2 = a^2\cos{2\theta} $$ And I am looking to find the length $x$ when $r=max$ Judging from the equation: $$r = \sqrt{a^2\cos{2\theta}} $$ R will be maximum ...
1
vote
1answer
338 views

Find the area of the region determined by two curves

Find the area of the region $R$ given by two curves. So the region $R$ describes the area that is common between the two curves: $$\begin{align*} \text{Function 1: } r&= 2\sin(\theta)\\ ...
5
votes
2answers
1k views

Is $r=2\cos(\theta)$ a one-petal polar function?

I'm currently learning about polar functions and their graphs in precalculus, and one of the questions on my homework is to identify the shape of the function $r=2\cos(\theta)$. We were taught that ...
1
vote
1answer
817 views

Replace the Cartesian equation $(x-5)^2 + y^2 = 25$ by an equivalent polar equation.

Replace the Cartesian equation $(x-5)^2 + y^2 = 25$ by an equivalent polar equation. Let $t= \theta$, $r=5$, $x=r\cos t$, $y=r\sin t$. I began with $x=5\cos t-5=5(\cos t-1)$ and $y=5\sin t$. Is that ...
0
votes
2answers
115 views

Laplacian in polar coordinates

I am stuck with an exercise that requires me to find the Laplacian $\Delta u=(D_x^2u+D_y^2u)$ of a 2d-function $u$ in polar coordinates (in the standard Euclidean plane). I found the following ...
0
votes
1answer
363 views

Transform integral into polar coordinates

At university we are given a voluntary hand in in the use of maple/matlab, in that regard I have a double integral I am in dire need to compute, using first cartesian then polarcoordinates. ...
2
votes
2answers
122 views

Polar Coordinates

It's been ages since i did any coordinate conversions, and typically i have these two which i just can't manage to solve by myself. I want to express the circle $x^{2}+y^{2}<4, x<0 $ The Area: ...
1
vote
1answer
271 views

Help needed with partial derivatives and polar coordinates, missing term.

I have a missing $\frac{1}{r}\partial_r$ -term (notice the question mark) but cannot see why, could someone hint where I am doing mistake.
2
votes
1answer
180 views

Explain Dot product with Partial derivatives in Polar-coordinates

Related to page 819 prob 4 in this book. I am incorrectly calculating the left-hand-side (def. LHS), some newbie error with commutativity probably. Ideas? Errors? I propose ...
2
votes
0answers
818 views

Explain Triangle perimeter in polar coordinates

The question is to give a formula in $x$ and $y$ that gives all three sides of an equilateral triangle. The formula should not be true for points that are not part of the perimeter of the triangle. ...