Introduction to Calculating the Volume of a Solid of Revolution

Understanding the volume of a solid generated by rotating a given region is crucial in many fields, including engineering and physics. In this article, we will explore the method of washers to find the volume of a solid. Specifically, we'll calculate the volume of the solid generated by rotating the region bounded by the curves y e^x, y 1, and x 2 about the x-axis. This process will be detailed step-by-step, ensuring that each concept is clearly explained and easy to follow.

Step 1: Identifying the Boundaries

To solve the volume problem, we first need to identify the boundaries of the region we are interested in. The region is bounded by the curves y e^x, y 1, and x 2. The curve y e^x intersects y 1 when e^x 1, which occurs at x 0. Therefore, the region of interest is between x 0 and x 2.

Step 2: Setting Up the Volume Integral

The volume of the solid of revolution can be found using the method of washers. This method involves finding the area of circular disks or washers at each point along the axis of rotation and summing these areas to find the total volume.

The Outer Radius and Inner Radius

In this case, the outer radius R is given by the curve y e^x, and the inner radius r is given by the line y 1.

Step 3: Defining the Integral

Using the method of washers, the volume is given by the integral:

V pi int_a^b (R^2 - r^2) dx

Substituting the outer and inner radii:

V pi int_0^2 (e^{2x} - 1) dx

Step 4: Calculating the Integral

To find the volume, we need to evaluate the integral:

V pi left[ frac{e^{2x}}{2} - x right]_0^2

Calculating this at the boundaries:

At x 2: frac{e^{2 cdot 2}}{2} - 2 frac{e^4}{2} - 2

At x 0: frac{e^{2 cdot 0}}{2} - 0 frac{1}{2}

Putting it all together:

V pi left(frac{e^4}{2} - 2 - frac{1}{2} right) pi left(frac{e^4}{2} - frac{5}{2} right)

To simplify, we get:

V frac{pi}{2} e^4 - 5

Conclusion

Therefore, the volume of the solid generated by rotating the region about the x-axis is:

V frac{pi}{2} e^4 - 5

Understanding the method of washers and the steps involved can help in solving a wide range of similar volume problems. If you are interested in other methods of rotation or specific axis rotations, please note that the region can also be rotated around the y-axis or other lines. The calculations and concepts would change accordingly, but the fundamental method of using integration remains the same.