Finding the Volume of the Solid

Oct 13, 2025 by windailys.com 102 views

Introduction

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) = e^x, the horizontal line y = 6, 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

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.

Finding the 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) = e^x, and the height of each disk is given by the difference between the horizontal line y = 6 and the function f(x) = e^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) = e^x, and the height of each disk is given by the difference between the horizontal line y = 6 and the function f(x) = e^x.

The radius of each disk is given by:

r = f(x) = e^x

The height of each disk is given by:

h = 6 - f(x) = 6 - e^x

Substituting the Values into the Formula

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

V = πr^2h

Substituting the values of r and h, we get:

V = π(e^x)2(6 - e^x)

Simplifying the Expression

To simplify the expression, we can expand the square and combine like terms:

V = π(e^2x)(6 - e^x)

Evaluating the Integral

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

V = ∫[0, ln(6)] π(e^2x)(6 - e^x) dx

Using Integration by Parts

To evaluate the integral, we can use integration by parts. Let u = e^2x and dv = (6 - e^x) dx. Then du = 2e^2x dx and v = -e^x + 6x.

Substituting the Values into the Integral

Now that we have found the values of u and v, we can substitute these values into the integral:

V = ∫[0, ln(6)] π(e^2x)(-ex + 6x) dx

Evaluating the Integral

To evaluate the integral, we can use the fundamental theorem of calculus. The integral is given by:

V = π[-(e^3x)/3 + 6(e^2x)/2] from 0 to ln(6)

Simplifying the Expression

To simplify the expression, we can evaluate the limits and combine like terms:

V = π[-(e^3ln(6))/3 + 6(e^2ln(6))/2]

Using the Properties of Exponents

To simplify the expression, we can use the properties of exponents. The expression can be simplified as follows:

V = π[-(6^3)/3 + 6(6^2)/2]

Evaluating the Expression

To evaluate the expression, we can simplify the fractions and combine like terms:

V = π[-216 + 108]

Simplifying the Expression

To simplify the expression, we can combine like terms:

V = π[-108]

The Final Answer

The final answer is:

V = 108π

Conclusion

In this article, we have found the volume of the solid generated by the region R, which is enclosed by the function f(x) = e^x, the horizontal line y = 6, and the y-axis. The volume of the solid is given by the formula V = 108π. This problem is a classic example of a solid of revolution, and the method of disks is used to find the volume of the solid.

References

[1] "Calculus" by Michael Spivak
[2] "Calculus: Early Transcendentals" by James Stewart
[3] "Mathematics for the Nonmathematician" by Morris Kline

Q: What is the region R, and how is it defined?

A: The region R is the area enclosed by the function f(x) = e^x, the horizontal line y = 6, and the y-axis. It is a solid of revolution, where the region R is rotated about the y-axis to form a solid.

Q: What is the method of disks, and how is it used to find the volume of the solid?

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.

Q: How do you calculate the volume of each disk?

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) = e^x, and the height of each disk is given by the difference between the horizontal line y = 6 and the function f(x) = e^x.

Q: What is the formula for the volume of each disk?

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.

Q: How do you simplify the expression for the volume of each disk?

A: To simplify the expression for the volume of each disk, you can expand the square and combine like terms. This will give you a simplified expression for the volume of each disk.

Q: How do you evaluate the integral to find the total volume of the solid?

A: To evaluate the integral, you can use the fundamental theorem of calculus. This involves evaluating the limits and combining like terms to find the total volume of the solid.

Q: What is the final answer for the volume of the solid generated by the region R?

A: The final answer for the volume of the solid generated by the region R is V = 108π.

Q: What are some common applications of the method of disks?

A: The method of disks is commonly used to find the volume of solids of revolution, such as spheres, cylinders, and cones. It is also used in engineering and physics to find the volume of complex shapes.

Q: What are some common mistakes to avoid when using the method of disks?

A: Some common mistakes to avoid when using the method of disks include:

Failing to identify the axis of rotation
Failing to calculate the radius and height of each disk
Failing to simplify the expression for the volume of each disk
Failing to evaluate the integral correctly

Q: How can I practice using the method of disks?

A: You can practice using the method of disks by working through examples and exercises in your textbook or online resources. You can also try using the method of disks to find the volume of different shapes and solids.

Q: What are some additional resources for learning about the method of disks?

A: Some additional resources for learning about the method of disks include:

Online tutorials and videos
Textbooks and workbooks
Online forums and discussion groups
Calculus courses and workshops