Calculating the Volume of Each Disk

Oct 13, 2025 by windailys.com 37 views

Let the Region R be Enclosed by the Function f(x) = 2√x, the Horizontal Line y = 2, and the y-axis: Finding the Volume of the Solid Generated

In this article, we will explore the problem of finding the volume of a solid generated by the region R, which is enclosed by the function f(x) = 2√x, the horizontal line y = 2, and the y-axis. This problem is a classic example of a solid of revolution, where the region R is rotated about the y-axis to form a solid. We will use the method of disks to find the volume of the solid.

The method of disks is a technique used to find the volume of a solid of revolution. It involves dividing the region R into thin disks, each of which is perpendicular to the axis of rotation. The volume of each disk is then calculated, and the sum of the volumes of all the disks is used to find the total volume of the solid.

To find the volume of the solid, we need to calculate the volume of each disk and then sum up the volumes of all the disks. The volume of each disk is given by the formula:

V = πr^2h

where r is the radius of the disk and h is the height of the disk.

In this case, the radius of each disk is given by the function f(x) = 2√x, and the height of each disk is given by the difference between the horizontal line y = 2 and the function f(x) = 2√x.

Calculating the Volume of Each Disk

To calculate the volume of each disk, we need to find the radius and height of each disk. The radius of each disk is given by the function f(x) = 2√x, and the height of each disk is given by the difference between the horizontal line y = 2 and the function f(x) = 2√x.

The radius of each disk is given by:

r = 2√x

The height of each disk is given by:

h = 2 - 2√x

Substituting the Values into the Formula

Now that we have the radius and height of each disk, we can substitute these values into the formula for the volume of each disk:

V = π(2√x)^2(2 - 2√x)

Simplifying the expression, we get:

V = π(4x)(2 - 2√x)

Evaluating the Integral

To find the total volume of the solid, we need to integrate the volume of each disk with respect to x. The integral is given by:

V = ∫[0,1] π(4x)(2 - 2√x) dx

Evaluating the integral, we get:

V = π(2 - 2√x)^2

Simplifying the Expression

Simplifying the expression, we get:

V = π(4 - 8√x + 4x)

Finding the Final Answer

To find the final answer, we need to evaluate the expression at the limits of integration. The limits of integration are from x = 0 to x = 1.

Evaluating the expression at x = 1, we get:

V = π(4 - 8 + 4)

Simplifying the expression, we get:

V = π(0)

However, this is not the correct answer. We need to evaluate the expression at x = 0.

Evaluating the expression at x = 0, we get:

V = π(4 - 0 + 0)

Simplifying the expression, we get:

V = 4π

In this article, we have found the volume of the solid generated by the region R, which is enclosed by the function f(x) = 2√x, the horizontal line y = 2, and the y-axis. The volume of the solid is given by the formula:

V = 4π

This is the final answer to the problem.

[1] "Calculus: Early Transcendentals" by James Stewart
[2] "Calculus" by Michael Spivak
[3] "Mathematics for Computer Science" by Eric Lehman and Tom Leighton

The problem of finding the volume of a solid of revolution is a classic example of a solid of revolution. The method of disks is a technique used to find the volume of a solid of revolution. The volume of each disk is given by the formula:

V = πr^2h

where r is the radius of the disk and h is the height of the disk.

In this case, the radius of each disk is given by the function f(x) = 2√x, and the height of each disk is given by the difference between the horizontal line y = 2 and the function f(x) = 2√x.

The volume of each disk is given by:

V = π(2√x)^2(2 - 2√x)

Simplifying the expression, we get:

V = π(4x)(2 - 2√x)

The total volume of the solid is given by the integral:

V = ∫[0,1] π(4x)(2 - 2√x) dx

Evaluating the integral, we get:

V = π(2 - 2√x)^2

Simplifying the expression, we get:

V = π(4 - 8√x + 4x)

Evaluating the expression at x = 1, we get:

V = π(4 - 8 + 4)

Simplifying the expression, we get:

V = π(0)

However, this is not the correct answer. We need to evaluate the expression at x = 0.

Evaluating the expression at x = 0, we get:

V = π(4 - 0 + 0)

Simplifying the expression, we get:

V = 4π

This is the final answer to the problem.
Q&A: Finding the Volume of the Solid Generated by the Region R

A: The region R is the area enclosed by the function f(x) = 2√x, the horizontal line y = 2, and the y-axis.

A: The method of disks is a technique used to find the volume of a solid of revolution. It involves dividing the region R into thin disks, each of which is perpendicular to the axis of rotation. The volume of each disk is then calculated, and the sum of the volumes of all the disks is used to find the total volume of the solid.

A: To calculate the volume of each disk, you need to find the radius and height of each disk. The radius of each disk is given by the function f(x) = 2√x, and the height of each disk is given by the difference between the horizontal line y = 2 and the function f(x) = 2√x.

A: The formula for the volume of each disk is:

V = πr^2h

where r is the radius of the disk and h is the height of the disk.

A: To simplify the expression for the volume of each disk, you need to substitute the values of r and h into the formula and then simplify the resulting expression.

A: The total volume of the solid is given by the integral:

V = ∫[0,1] π(4x)(2 - 2√x) dx

A: To evaluate the integral, you need to integrate the expression π(4x)(2 - 2√x) with respect to x from 0 to 1.

A: The final answer to the problem is:

V = 4π

A: The region R is a classic example of a solid of revolution, and the method of disks is a technique used to find the volume of a solid of revolution.

A: The method of disks has many common applications in physics, engineering, and mathematics, including finding the volume of a solid of revolution, calculating the surface area of a solid, and determining the moment of inertia of a solid.

A: Some tips for solving problems involving the method of disks include:

Make sure to understand the problem and the method of disks before starting to solve it.
Use the formula V = πr^2h to calculate the volume of each disk.
Simplify the expression for the volume of each disk by substituting the values of r and h into the formula.
Evaluate the integral to find the total volume of the solid.
Check your work to make sure that you have found the correct answer.

A: Some common mistakes to avoid when solving problems involving the method of disks include:

Failing to understand the problem and the method of disks.
Making errors when calculating the volume of each disk.
Failing to simplify the expression for the volume of each disk.
Making errors when evaluating the integral.
Failing to check your work to make sure that you have found the correct answer.