00:01
All right, our first problem.
00:02
This is two problems, and each problem has two parts.
00:05
We're given y equals the square root of x, and we're bound by y equals 2 and x equals 0, i believe.
00:21
X equals 0, y equals 2 and x equals 0, yeah.
00:27
All right.
00:28
So draw this really quick.
00:32
We want x equals 0, which is our y -axis.
00:35
And y equals 2.
00:42
So that's going to be our line y equals 2.
00:44
Draw a curve.
00:49
And we intersect that curve out here at x equals 4.
00:55
All right.
00:57
So this is the region we are trying to rotate right here.
01:08
And our first part a is rotating around the x -axis, or sorry, the y -axis, which isn't too bad.
01:18
Our radius as we rotate is going to be from 0 to 4.
01:23
So our integral will be from 0 to 4, pi times our function square root of x squared dx.
01:35
Go ahead and solve this really quickly.
01:38
Move our pi out front.
01:40
From 0 to 4 of x, dx.
01:44
And this just gives us a final answer of 8 pi.
01:48
For part b, we're rotating around x equals 4.
01:53
So we're actually rotating at this line here.
01:57
Let me put that in a different color, just so it is a little clearer.
02:02
That is our second x equals 4, and our first was here at x equals 0 or the y -axis.
02:18
So come over here, we're going to have to go ahead and find our outside radius, which is going to be from here to here.
02:30
That is the longest radius we have, so it'll be the outside radius.
02:34
And that is going to be four units.
02:42
Is our or.
02:43
Our inside radius, well, that's going to be our curve minus this line.
02:51
So it's going to be y minus four, which is equal to the square root of x minus four.
03:02
Now set up our integrals.
03:04
We got from zero to four, from zero to four, pi times.
03:16
Our radius squared, and it's our outside radius, so 4 squared, dx minus 0 to 4, pi times radical x minus 4 squared dx.
03:36
Go ahead and type this into a calculator after you pull out our pi.
03:39
So what you would be typing into your calculator, i'm going to pull a pie out here, and i'll write in white what we're typing into our calculator.
03:50
Then it goes from 0 to 4, 4 squared dx, or you just type 16, that's fine too, minus from 0 to 4, square root of x minus 4 squared dx.
04:08
And then at the end, we're going to multiply that by pi.
04:11
Typing this into our calculator, we end up getting, one second to type it in, we get from 0 to 4 of 16 dx minus, from 0 to 4 of the square root of x minus 4 squared d x that gives us a answer of pi times 34 .6 repeating, which you can use your calculator to convert that into a fraction...