00:01
All right in this question we are asked to find the volume of the region bounded by a low by a low bounded is bounded below the plane right this plane there a set is equal to zero latered by the cycular cylinder and this is the circular cylinder there right right, it's laterally by the circular cylinder there and above by the paraboloid there, the paraboloid.
00:47
Okay, so now let me write this is a sketch that we have our z there, right, and then no i know it, so we're trying to come up with that plane there.
01:05
So i'm sure the sketch we see there, plane above there, right? then this side and this side, okay, so you're saying z here is equal to r at theta is equal to zero, right? right, right.
01:22
Then z is going to r at theta is supposed to pi over two there.
01:28
Then theta is equal to zero as you go towards pi over two there.
01:37
Okay.
01:40
The point there as z is equal to 2 there okay right so now if we just so here i just put the inequalities to come up with the limits so my z must be greater than or equal to zero there but it must be less than x plus x squared plus y squared there and then my x squared plus y, the circular cylinder there must be at least less than one day.
02:23
And then our paraboloid there must be at least less than or equal to.
02:32
So these are the inequalities that we can state there.
02:37
Right.
02:37
We're trying to make sense out of the question so that when we answer it properly, when we we will do it properly okay so now let me just transform this so here we are transforming this the transforming this lens transforming them to cylindrical coordinates there so that we can so so we are transforming the all these right to cylindrical coordinates there right so we know our theta there it must be greater than or equal to zero the part at the same time less than pi there okay right and our radius r we know the radius must be equal to less than or equal to sign you know the radius of a cylinder's two sign theta the part must be it is greater than zero there and then our z there must be greater than 0 but at least less than r squared there.
03:57
All right, it must be less than or equal to r squared there.
04:02
Okay.
04:03
Right, let's go on to our triple regression there to find the volume there.
04:15
So, right, if we are going to, if we do.
04:20
Our triple integration there to find the volume it will be the interval of dx, then d z so in this three dimensional way okay so that one is transformed to when we have cylindrical coordinates it is now transformed to right uh -huh we said our right t theta there our theta must be from zero to pi and our r from 0 to 2 sine theta there then inside there we have right our z there our z we'll start from 0 to r squared there okay so we have r dz there okay so now let's just integrate this we do triple integration so we'll start from inside there so we're just integrating this so if we integrate this what we get we get r z there so rz there right where there is it we put r squared so we come up with r cubed then because z we replace with rr squared there so that's why our squared there that's why our r squared there mean r cubed so r cubed tera the theta right from sine from zero to sign to sign to sign to sign theta all right right um it's clear there now let's differentiate uh with respect to r there so our d r right now we are integrating not differentiating we are integrating right according to the r there it becomes error to the power of four over four there right arrow to the power of four over 4, right, from 0 to 0 there to 2 sine theta day.
06:36
So 2 sine theta will replace r there...