25 points scalc9m 157025ep conssder the following the...

Name: 25 points scalc9m 157025ep conssder the following the wolume of the solid that is enclosed by the cone z sqrtx2 y2 and the sphere x2 y2 z2 162 using cylindrical coordinates write an integra 08703
Uploaded: 2023-09-22T18:31:45-08:00
Duration: 8 min 1 s
Channel: William Mead
Description: 25 points scalc9m 157025ep conssder the following the wolume of the solid that is enclosed by the cone z sqrtx2 y2 and the sphere x2 y2 z2 162 using cylindrical coordinates write an integra 08703

[-/2.5 Points] ◻ ◻ SCALC9M 15.7.025.EP. Conssder the following. the wolume of the solid that is enclosed by the cone z-\sqrt(x^(2) y^(2)) and the sphere x^(2) y^(2) z^(2)-162 Using cylindrical coordinates, write an integral that can be evaluated to find the volume V of the given solid. (Choose V=\int_0^A \int_0^B \int_r^C rdrdrd\theta ◻◻c=◻◻◻r.0. Choose r.0. Choose 0. Choose r.

Transcript

00:01 I'd like to find the volume of this compound shape here.

00:03 It's got a cone on top, where we've got z is equal to 2 minus the square root of x squared plus y squared.

00:13 That is equal to z.

00:14 Or z is equal to x squared plus y squared.

00:20 To find the point where they intersect, that is the circle there, we just set these guys equal to each other.

00:26 And it hints us to find the radius of the circle.

00:30 These are both very clearly rotationally symmetric around the z -axis.

00:35 So if we can think of it in cylindrical coordinates, x squared plus y squared is just r squared.

00:41 So 2 minus the square root of r squared is equal to r squared.

00:46 Square root of r squared is just r.

00:48 So we can solve here, r squared plus r minus 2 is equal to 0.

00:54 We can factor this as r plus 1 times r minus 2 is equal to 0.

01:00 1 times negative 2 is 2.

01:01 1 plus negative 2 is 1.

01:03 Oh, excuse me.

01:06 R minus 1 plus 2.

01:09 So that we get negative 1 times 2 is negative 2.

01:13 And 2 minus 1 is 1.

01:14 Great.

01:15 So r either has to equal 1 or r equals negative 2.

01:19 R obviously is not negative 2, because r is strictly positive.

01:22 So our radius is 1.

01:26 That's great and cool.

01:28 That means that along this circle here, we have z is equal to r squared is equal to 1 squared is equal to 1.

01:38 And likewise here, z is equal to 2 minus square root of r squared is 2 minus the square root of 1, which is 2 minus 1, which is 1.

01:50 So you've got that.

01:51 That's great.

01:53 And of course, in the middle here, we have z is equal to 2 minus the square root of 0, which is 2.

02:02 So this cone has radius equal to 1 and height equal to 1 as well, because we're going from 2, from z equals 2 down to z equals 1.

02:15 We're just going to have volume 1 third.

02:18 Oh, excuse me.

02:21 1 third, because it's a cone, times pi times the height, which is 1, times the radius squared, which is also 1, which is pi on 3 for the volume of the cone.

02:35 For the paraboloid, it's going to be a little bit trickier.

02:39 So i'm actually going to go ahead and set up a triple integral here in cylindrical coordinates.

02:45 Z is going from 0 to 1.

02:50 R is going from, let's see, 0, or what do we got here, actually? well, easily, theta is going from 0 to 2 pi.

02:59 So let's actually just pull out a factor of 2 pi, and then we'll see what we got here.

03:06 We'll say z goes from 0 to 1, and then the radius here, the radius squared is equal to z.

03:15 R squared is equal to z, which means r is the square root of z.

03:20 So we're actually going to go from 0 to the square root of z in general for r.

03:26 Our volume is just 1, so then we get r dr dz...

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