Given two surface equation, convert to polar coordinates and find the volume of the solid using triple integral

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Consider the surfaces given by
$$
M:(x^2+z^2-z)^2=(x^2+z^2)
$$
$$
N:y^2=x^2+z^2
$$

(a) Notice that since there is no y present in M the shape of the surface is a cylinder determined by its shape in the xz-plane! Convert M to polar coordinates and sketch the surface. What’s the name of the type of curve that determines its shape? Note: You may assume r not euqal 0.

(b) Sketch the solid bounded by M, N and the xz-plane.

(c) Find the volume of the solid by setting up an appropriate triple integral and computing.
Hint: Polar! Also, some functions are odd.

a) I got $$r=0$$ and $$r=1+sin theta $$

and i know the shape of r=1+sin theta is a cardioid

b) I know N is a elliptical cone, but I dont know how to sketch them on a graph, and I have no idea how the solid would look like.

I am stuck afterward...

calculus integration polar-coordinates volume

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asked Jul 24 at 5:14

gummug

1

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Consider the surfaces given by
$$
M:(x^2+z^2-z)^2=(x^2+z^2)
$$
$$
N:y^2=x^2+z^2
$$

(a) Notice that since there is no y present in M the shape of the surface is a cylinder determined by its shape in the xz-plane! Convert M to polar coordinates and sketch the surface. What’s the name of the type of curve that determines its shape? Note: You may assume r not euqal 0.

(b) Sketch the solid bounded by M, N and the xz-plane.

(c) Find the volume of the solid by setting up an appropriate triple integral and computing.
Hint: Polar! Also, some functions are odd.

a) I got $$r=0$$ and $$r=1+sin theta $$

and i know the shape of r=1+sin theta is a cardioid

b) I know N is a elliptical cone, but I dont know how to sketch them on a graph, and I have no idea how the solid would look like.

I am stuck afterward...

calculus integration polar-coordinates volume

share|cite|improve this question

asked Jul 24 at 5:14

gummug

1

add a comment | 

up vote
0
down vote

favorite

up vote
0
down vote

favorite

Consider the surfaces given by
$$
M:(x^2+z^2-z)^2=(x^2+z^2)
$$
$$
N:y^2=x^2+z^2
$$

(a) Notice that since there is no y present in M the shape of the surface is a cylinder determined by its shape in the xz-plane! Convert M to polar coordinates and sketch the surface. What’s the name of the type of curve that determines its shape? Note: You may assume r not euqal 0.

(b) Sketch the solid bounded by M, N and the xz-plane.

(c) Find the volume of the solid by setting up an appropriate triple integral and computing.
Hint: Polar! Also, some functions are odd.

a) I got $$r=0$$ and $$r=1+sin theta $$

and i know the shape of r=1+sin theta is a cardioid

b) I know N is a elliptical cone, but I dont know how to sketch them on a graph, and I have no idea how the solid would look like.

I am stuck afterward...

calculus integration polar-coordinates volume

share|cite|improve this question

asked Jul 24 at 5:14

gummug

1

Consider the surfaces given by
$$
M:(x^2+z^2-z)^2=(x^2+z^2)
$$
$$
N:y^2=x^2+z^2
$$

(a) Notice that since there is no y present in M the shape of the surface is a cylinder determined by its shape in the xz-plane! Convert M to polar coordinates and sketch the surface. What’s the name of the type of curve that determines its shape? Note: You may assume r not euqal 0.

(b) Sketch the solid bounded by M, N and the xz-plane.

(c) Find the volume of the solid by setting up an appropriate triple integral and computing.
Hint: Polar! Also, some functions are odd.

a) I got $$r=0$$ and $$r=1+sin theta $$

and i know the shape of r=1+sin theta is a cardioid

b) I know N is a elliptical cone, but I dont know how to sketch them on a graph, and I have no idea how the solid would look like.

I am stuck afterward...

calculus integration polar-coordinates volume

share|cite|improve this question

asked Jul 24 at 5:14

gummug

1

share|cite|improve this question

share|cite|improve this question

asked Jul 24 at 5:14

gummug

1

asked Jul 24 at 5:14

gummug

1

asked Jul 24 at 5:14

gummug

1

1

add a comment | 

add a comment | 

1 Answer
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$x = rcostheta\
z = rsin theta\
y= y$

Jacobain

$dx dy dz = r dy dr dtheta$

Regarding $M,$ as you say $r = 1 + sin theta$ which is a cartiod cylinder.
$N$ is a double cone (not elliptical)

$y^2 = r^2\
y = pm r$

$V = int_0^2piint_0^1-sinthetaint_-r^r r dy dr dtheta$

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answered Jul 24 at 5:31

Doug M

39.1k31749

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1 Answer
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1 Answer
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oldest

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up vote
1
down vote

$x = rcostheta\
z = rsin theta\
y= y$

Jacobain

$dx dy dz = r dy dr dtheta$

Regarding $M,$ as you say $r = 1 + sin theta$ which is a cartiod cylinder.
$N$ is a double cone (not elliptical)

$y^2 = r^2\
y = pm r$

$V = int_0^2piint_0^1-sinthetaint_-r^r r dy dr dtheta$

share|cite|improve this answer

answered Jul 24 at 5:31

Doug M

39.1k31749

add a comment | 

up vote
1
down vote

$x = rcostheta\
z = rsin theta\
y= y$

Jacobain

$dx dy dz = r dy dr dtheta$

Regarding $M,$ as you say $r = 1 + sin theta$ which is a cartiod cylinder.
$N$ is a double cone (not elliptical)

$y^2 = r^2\
y = pm r$

$V = int_0^2piint_0^1-sinthetaint_-r^r r dy dr dtheta$

share|cite|improve this answer

answered Jul 24 at 5:31

Doug M

39.1k31749

add a comment | 

up vote
1
down vote

up vote
1
down vote

$x = rcostheta\
z = rsin theta\
y= y$

Jacobain

$dx dy dz = r dy dr dtheta$

Regarding $M,$ as you say $r = 1 + sin theta$ which is a cartiod cylinder.
$N$ is a double cone (not elliptical)

$y^2 = r^2\
y = pm r$

$V = int_0^2piint_0^1-sinthetaint_-r^r r dy dr dtheta$

share|cite|improve this answer

answered Jul 24 at 5:31

Doug M

39.1k31749

$x = rcostheta\
z = rsin theta\
y= y$

Jacobain

$dx dy dz = r dy dr dtheta$

Regarding $M,$ as you say $r = 1 + sin theta$ which is a cartiod cylinder.
$N$ is a double cone (not elliptical)

$y^2 = r^2\
y = pm r$

$V = int_0^2piint_0^1-sinthetaint_-r^r r dy dr dtheta$

share|cite|improve this answer

answered Jul 24 at 5:31

Doug M

39.1k31749

share|cite|improve this answer

share|cite|improve this answer

answered Jul 24 at 5:31

Doug M

39.1k31749

answered Jul 24 at 5:31

Doug M

39.1k31749

answered Jul 24 at 5:31

Doug M

39.1k31749

39.1k31749

add a comment | 

add a comment | 

 

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