Question

parameterize the closed surface obtained by rotating the region inside the
first quadrant enclosed by
y = x^3
y = 4x
about the x-axis

Answer 1

Find their two points of intersection. Imagine the axes laid out, or even better draw a picture so you can label x, r, and theta since this is just cylindrical coordinates. I guess show me where you're getting stuck since I know you're smart and I am probably just repeating stuff you know lol.

Answer 2

i would use cylindrical coordinates as well, one moment

Answer 3

i learned how to deal with one surface but the two surfaces are confusing me at the moment

Answer 4

lets use 'modified' cylindrical coordinates
( r , theta, x )

Answer 5

does that work when the surface is made up of two surfaces ?

Answer 6

Just do them separately @ganeshie8. One is the outside shell and one is simply the inside shell. Just find out when the two functions (radii) intersect.

Answer 7

My bad i have lost my latex work :(

Answer 8

they intersect at (0, 0) and (2, 8)

Answer 9

Yep, so you can just parametrize both in cylindrical coordinates then define your domain to be [0,2] and you're done, that's it.

Answer 10

x = u
y = u^3cos(\theta)
z = u^3sin(\theta)

for rotating first curve i think

Answer 11

u = [0, 2]

Answer 12

we need a different parameterization for the other piece is it ?

cant a single parameterization for the entire closed surface ?

Answer 13

Nope, not that I know of. Only a few nice geometric surfaces can be parameterized with a single function because their coordinate system is nice and matches up. For instance, the sphere requires a top and bottom in rectangular coordinates since the square root function is single valued.

Answer 14

You could no doubt define a new coordinate basis where it was possible to only define a single function, however it would be even more terrible than just gluing the two together like this

Answer 15

makes sense ! il try paameterizing the line, it shouldnt be hard
i hope the same parameters \(r\) and \(\theta\) change smoothly across both the surfaces

because this is a single closed surface.. im just worried splitting the parameterization breaks anything..

Answer 16

this is a nice problem to go along with your last question on parametrization a surface of revolution

Answer 17

parametrizing*

Answer 18

The derivative around the "seam" where the two connect together is not differentiable if that's what you mean @ganeshie8 It's just like a piecewise continuous function or differentiating absolute value.

Answer 19

thanks the two pieces are working perfectly !

Answer 20

what was your final expression, can you post it ?

Answer 21

i can try to graph it on my graphing calculator program , which can handle parametrically defined surfaces

Answer 22

perl i made two different parameterization as discussed above

y=x^3 :
```
x = r
y = r^3cos(\theta)
z = r^3sin(\theta)
```

y=4x:
```
x = r
y = 4rcos(\theta)
z = 4rsin(\theta)
```

r : [0, 2]
theta : [0, 2pi]

Answer 23

thanks