CoCalc Public FilesChapter 16 / Parametric Surfaces Solutions.sagewsOpen with one click!
Author: John Hoggard
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Description: Some solutions to the parametric surface exercises from EUP Calc III, Spring 2016 (Hoggard)
Compute Environment: Ubuntu 18.04 (Deprecated)

Solutions: Parametric Exercises

  1. The top half of a hemisphere of radius 3.

Easiest is probably to use polar (or something equivalent):

var("r, t") parametric_plot3d((r*cos(t), r*sin(t), sqrt(9 - r^2)), (r, 0, 3), (t, 0, 2*pi))
(r, t)
3D rendering not yet implemented

But, there's no reason why we couldn't let the xx-coordinates range between ±3\pm 3, while the yy coordinates need to go between ±9x2\pm \sqrt{9 - x^2}:

var("u, v") parametric_plot3d((u, v*sqrt(9-u^2), sqrt(9-u^2-(v^2*(9-u^2)))), (u, -3, 3), (v, -1, 1))
(u, v)
3D rendering not yet implemented

(In short, there are many parametric representations of the same surface.)

  1. The surface z=x+2yz = x + 2 y over the region where 0yx20 \leq y \leq x^2, 0x10 \leq x \leq 1. Then see if you can add the sides to make a picture of the region under consideration when computing 010x2x+2y,dy,dx \int_0^1 \int_0^{x^2} x + 2 y , dy , dx from section 15.3.

Solution:

var("u, v") regtop = parametric_plot3d((u, v*u^2, (u + 2*v*u^2)), (u, 0, 1), (v, 0, 1), color="darkgreen") regfront = parametric_plot3d((1, u, v*(1+2*u)), (u, 0, 1), (v, 0, 1), color="olive") regleft = parametric_plot3d((u, 0, v*(u+0)), (u, 0, 1), (v, 0, 1), color="olivedrab") regback = parametric_plot3d((u, u^2, v*(u + 2*u^2)), (u, 0, 1), (v, 0, 1), color="green") regtop + regfront + regleft + regback
(u, v)
3D rendering not yet implemented
  1. The region trapped between z=x2+3y2z = x^2 + 3 y^2 and z=8x2y2z = 8 - x^2 - y^2, which we found the volume of in section 15.10.

Solution: Start by finding the intersection of these curves. We have x2+3y2=8x2y2 x^2 + 3 y^2 = 8 - x^2 - y^2 which gives the ellipse in the xyxy plane x24+y22=1 \frac{x^2}{4} + \frac{y^2}{2} = 1 We can parametrize these with x=2cos(θ)x = 2 \cos(\theta) and y=2sin(θ)y = \sqrt{2} \sin(\theta), so our top and bottom should look like this:

var("r, t") solidtop = parametric_plot3d((2*r*cos(t), sqrt(2)*r*sin(t), 8 - (2*r*cos(t))^2 - (sqrt(2)*r*sin(t))^2), (r, 0, 1), (t, 0, 2*pi), color="tan") solidbot = parametric_plot3d((2*r*cos(t), sqrt(2)*r*sin(t), (2*r*cos(t))^2 +3*(sqrt(2)*r*sin(t))^2), (r, 0, 1), (t, 0, 2*pi), color="coral") solidtop+solidbot
(r, t)
3D rendering not yet implemented
  1. On Test 3, problem 1(c) asked you to find the region being integrated over for x=03y=09x2z=03xf(x,y,z),dz,dy,dx \int_{x=0}^3 \int_{y=0}^{9-x^2} \int_{z = 0}^{3-x} f(x, y, z) , dz , dy , dx (It was choice (G) of the options presented, and is shown below.)

Solution:

var("u, v") testgtop = parametric_plot3d((u, v*(9-u^2), (3-u)), (u, 0, 3), (v, 0, 1)) testgfront = parametric_plot3d((u, 9-u^2, v*(3-u)), (u, 0, 3), (v, 0, 1), color="teal") testgback = parametric_plot3d((0, u, v), (u, 0, 9), (v, 0, 3), color="steelblue") # Of course, you can't see this side on the test... testgleft = parametric_plot3d((u, 0, v*(3-u)), (u, 0, 3), (v, 0, 1), color="navy") # Or this side either! testgtop + testgfront + testgback + testgleft
(u, v)
3D rendering not yet implemented

In fact, I've left bottoms off of all of these, but they would not be hard to add if you wished.

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