Google Classroom
Gain some risk-free experience with applying the disc method in this worksheet, before you attempt our exercise.
Problem 1
A region is enclosed by the
What is the volume of the solid generated when this region is rotated around the
If we rotate the enclosed region about the
The image below shows a disk cross section of the solid.
The radius of each disk is
The volume of each disk can be found by multiplying the area of the face by the width of the disk,
We want to sum up the volumes of infinitely many of these disks between
We can evaluate the definite integral to find the volume of the solid.
The volume of the solid is
Problem 2
A region is enclosed by the
What is the volume of the solid generated when this region is rotated around the
This problem is similar to problem 1, except we aren't specifically given the left and right endpoints of the enclosed region. To find these, we need to find where the curve
A sketch of the given region and one representative disk is shown below.
We see that the radius of each disk is
To find the volume, we multiply the area of the face by the width of the disk,
We want to sum up the volumes of infinitely many of these disks between
We can evaluate the definite integral to find the volume of the solid.
The volume of the solid is
Problem 3
A region is enclosed by the
What is the volume of the solid generated when this region is rotated around the line
In this problem, the enclosed region is not rotated about the
Let's sketch this region and a cross section of the solid. Notice that
Now we can see that the radius of each disk is
The definite integral
We evaluate the integral below.
The volume of the solid is
Problem 4
A region is enclosed by the positive
What is the volume of the solid generated when this region is rotated around the
In this problem, note that the enclosed region is rotated about the
Let's sketch this region and a cross section of the solid.
Here the disks are horizontal, and so the width of each disk is
From the image, we see that the radius of each disk is
We can now express the area of the face of each disk as
To find the volume of the entire solid, we sum up the volumes of infinitely many disks between
We can evaluate the definite integral to find the volume of the solid.
The volume of the solid is
Problem 5
A region is enclosed by the
What is the volume of the solid generated when this region is rotated around the
We start by sketching the enclosed region and a representative cross section of our solid.
Here, the disks are horizontal, and so the width of each disk is
The radius of each disk is
So it follows that the area of a face of one disk is
We want to sum up the volumes of infinitely many of these disks between
We can evaluate the definite integral to find the volume of the solid.
The volume of the solid is
Problem 6
A region is enclosed by the
What is the volume of the solid generated when this region is rotated around the line
In this problem, the enclosed region is not rotated about the
Let's sketch this region and a cross section of the solid.
Notice that the disks are horizontal, and so the width of each disk is
Since
From the image, we see that the radius of a disk is
So the area of the face of each disk is
We want to sum up the volumes of infinitely many of these disks between
We can evaluate the definite integral to find the volume of the solid.
The volume of the solid is
Log in Leo Bekerman 8 years agoPosted 8 years ago. Direct link to Leo Bekerman's post “For problem 6, isn't the ...” For problem 6, isn't the radius of the disc (9 - y), not (9 - y^2)? The explanation says you can "see this from the graph" but what I see is [9 - f(x)] and f(x) = Y; therefore, r = (9 - y), right? • (3 votes) Bea 8 years agoPosted 8 years ago. Direct link to Bea's post “f(x)=√x, not y. You shoul...” f(x)=√x, not y. You should be integrating π∫[9-f(y)]dy. The given curve is y=√x, but if we're going to evaluate an integral using dy then our equation also has to be in terms of y. If we square both sides of y=√x, we get y²=x. So f(y)=y², and our radius r=9-f(y) is 9-y² hope this helps ♥ (8 votes) Brenden Delong 6 years agoPosted 6 years ago. Direct link to Brenden Delong's post “consider this problem "y=...” consider this problem "y=4-x^2 bounded by y=0 rotated around y=4 What is the volume?" • (2 votes) Liz Pulley 5 years agoPosted 5 years ago. Direct link to Liz Pulley's post “why are the bounds y=0 an...” why are the bounds y=0 and y=3.... how did you find y=3 • (1 vote) BreAnna 7 years agoPosted 7 years ago. Direct link to BreAnna's post “For # 6, why is the radiu...” For # 6, why is the radius of the disk (9-x) instead of (9-sqrtx) ? • (0 votes)Want to join the conversation?