00:01
So we're given this question, graph the shape formed by the graphs of x equals cosine of x squared and y equals sine of x squared.
00:11
Then find the volume of the solid generated by revolving this shape around the y axis.
00:17
So there's a few common steps involved with volumes of revolution.
00:21
And the first of those is graphing the given region.
00:24
So i've gone ahead and used an online graphing utility to do this.
00:27
X equals 0, y equals cosine of x squared, and y equals sine of x squared.
00:33
And it's also important to identify the axis of revolution.
00:38
Notice in the problem, it says that we're revolving the shape around the y -axis.
00:45
I'll go ahead and denote that on our graph.
00:49
That is, so the y -axis is given by the function that they tell us is x -equal zero.
00:55
So we're using y -axis revolution.
01:04
Okay.
01:05
So the third step is to then draw a rectangle that touches all y equals f of x functions.
01:14
And then after we've drawn that rectangle, determine the best method to use.
01:20
So i've gone ahead and drawn this rectangle in orange on our graph.
01:24
And then based on that rectangle, what it's touching, and where it's placed in position to our axis of revolution, we can determine which method is best to use.
01:33
So i've gone ahead and i've typed out the three methods, that are typically used for volumes of revolution.
01:41
The first of these methods is known as the disk method.
01:46
Now this is useful when the rectangle you've drawn is perpendicular to the axis of revolution.
01:54
And this is also true for the washer method, when the rectangle is perpendicular to the axis of revolution.
02:01
But notice that the rectangle we've drawn is parallel to the axis of revolution given to us in this problem.
02:08
So instead, we're going to use what's known as the shell method, which i've written here in orange.
02:14
So the shell method is used when the rectangle is parallel to the axis of revolution, and the rectangle also must touch both curves.
02:23
So notice that is what we have here.
02:26
So the formula for the volume of a shell is given here.
02:31
It's the integral from a to b of 2 pi rht.
02:36
So right now i'm going to go ahead and kind of describe what all of those components mean.
02:40
So let's start with a and b.
02:43
A and b are our bounds, kind of like when you're working with finding the area of a region.
02:50
It's the same idea.
02:52
So this is going to be our a and this is going to be our b.
03:01
So let me choose another color here really quick.
03:04
So this entire region is what we want to revolve around the y axis, all of this stuff.
03:24
So a is clearly seen as zero.
03:33
But b, this x value, we have to figure out what that x value is.
03:39
And we do that by finding the intersection point of our two curves.
03:46
So let's go over here and figure that out.
03:49
I'll write this in blue.
03:53
So we have y equals cosine of x squared, and we have y equals sine of x squared.
04:05
And so finding the intersection of both curves, we have to set these two functions equal to one another.
04:13
Now this problem looks a little confusing to solve, but let's really quick go back to what sign and cosine really mean.
04:20
So if you're finding the cosine of an angle, you're really looking at an x value over the hypotenus.
04:29
And this is in relation to the unit circle.
04:37
So and then sine is equal to our y component over our h.
04:45
So if we're looking at a triangle in the unit circle, this is our h, which is always equal to one when looking at the unit circle.
04:52
This is our x and this is our y.
04:55
So if we want our cosine and our sign to be equal to one another, like this part right here, cosine theta equals sine theta, which we know these now equal to x over h equals y over h.
05:14
That was an ugly age.
05:17
Then we know that we're looking for an angle, theta, where x is equal to y.
05:26
And so this is consistent with when theta is 45 degrees, or similarly, pi over 4 on the unit circle.
05:38
So let's extend this line up here.
05:42
So we now know that cosine of any angle is equal to sign of any angle when that angle is pi over 4.
05:51
So we know that cosine of pi over four is equal to sine of pi over four.
05:58
Awesome.
06:00
Well, what we've done is we've made a substitution for x squared equaling pi over four, because this is our angle.
06:08
So if we go ahead and solve this equation for x by taking the square root of both sides, we get that x is equal to the square root of pi over two.
06:19
And that becomes this x value here, this b value.
06:34
So now we have to, we have to the square root of pi over two.
06:34
And that becomes this b value.
06:34
So now we have to, have our bounds where a is equal to zero, b is equal to root pi over two.
06:49
All right, so let me scroll a little bit here.
06:54
I will rewrite the formula that we're working with.
06:59
The volume of a shell is the integral from a to b to pi, r -h -t, where we know that our a is zero, and our b is root pi over two.
07:21
Now we want to determine our r, our h, and our t.
07:26
So let's scroll back up to our curve for a second.
07:33
R is our radius.
07:37
So when this curve is revolved around the x, is revolved around the y axis this way, our radius is going to be this distance here.
07:48
Our radius is going to be an x value, right? because this is going to go moving in the x direction.
07:56
Our h is the height of our rectangle, which is an ever -changing y value, because this yellow rectangle, orange rectangle, i'm sorry, represents the changing values of x and y as it moves along this region.
08:19
Our t value is always going to be this little base, the size of this base of our rectangle, which is also ever -changing, which is given to us in this.
08:32
This differential piece, delta x.
08:36
So let's go back down here and plug in these ideas.
08:42
Our radius is an x component.
08:45
Our height is measured in the y direction...