00:01
All right, for your question, you are given a shape drawn on a coordinate system, and it's a trapezoid.
00:07
And we're supposed to find the volume of the solid formed by rotating the trapezoid around the x -axis.
00:14
So what you're really thinking is you're trying to rotate this around and understand what shape you're dealing with here.
00:22
It's not a common geometric shape, so there's not a specific formula to find it all at once.
00:28
What i've done is analyze this question and said this portion that i'll highlight in yellow.
00:38
If i rotated just that portion around, again, the outer edge of this shape is going to follow a circular path as it rotates.
00:49
Now, this is very hard to draw, so i'm not going to really attempt to draw this.
00:53
We'll attempt it.
00:54
But it's going to follow a circular shape around the x -act.
00:59
And what you should realize is that yellow portion is going to form a cone.
01:05
And we can find the volume of a cone.
01:10
Now, the formula for the volume of a cone is 1 3rd pi r squared h.
01:19
Okay, so for that cone, the radius would be here, the height would be here, so it's going to be 1 3rd times pi times the the radius would be the y coordinate.
01:33
So it's three squared times h, which looks to be the distance between the origin and this point here, which would be four.
01:44
Now, i can simplify this a little bit.
01:46
It'll make our lives a little easier.
01:49
Three squared is nine, and nine times one -third.
01:55
So if i square that, i get nine, and multiply it to one -third, i'm going to get three.
01:59
That 3 multiplies to the 4 and gives us 12 pi.
02:06
So the volume of the yellow region will be 12 pi.
02:10
We'll take it to a calculator when we're all done...