Mathematics Asked by 2316354654 on March 2, 2021
Sketch the solid described by the following inequalities.
I’m not sure how to aproach this. I read the section on cylindrical coordinates, and I know how to plot points in cylindrical, and switch from rectangle to polar, but I don’t know what to do with this. It seems like plotting points would take forever, and there should be some faster way.
It's a good idea to have some intuition on the shapes of common functions in cylindrical coordinates.
One thing I find the most helpful is equations relating $r$ and $z$. Since there is no $theta$ relation, the functions $z = z(r)$ all describe rotationally symmetric surfaces. The idea is if you can plot $z$ vs. $r$ in 2D, the 3D result is that curve revolved around the $z$ axis. If you've done anything similar to surfaces and volumes of revolution in single-variable calculus, this is the same idea. F
or example, functions of the form
$$ z = ar^2 + b $$
Looks like a parabola on the plane, so in 3D it is a paraboloid
Functions of the form $$ z = ar $$
are diagonal lines revolved around the $z$-axis, so they're shaped like a cone.
When these functions are combined with limits on $theta$ (like in your second example), it restricts the angle of revolution. So $0 le theta le pi/2$ on makes a quarter revolution in the first quadrant, and in effect looks like quarter cone.
Answered by Dylan on March 2, 2021
Get help from others!
Recent Questions
Recent Answers
© 2024 TransWikia.com. All rights reserved. Sites we Love: PCI Database, UKBizDB, Menu Kuliner, Sharing RPP