AP Calculus AB: Lesson 7.4: Disk and Washer Method

This paragraph introduces the topic of calculating the volume of solids created by revolving a region around an axis using definite integrals. The focus is on two methods: the disk method and the washer method. The disk method involves finding the volume of a solid obtained by rotating a region around the x-axis, starting with a simple rectangle and considering its transformation into a circular disc when rotated. Key terms and the foundational concept of revolving a region to form a solid figure are explained, setting the stage for deeper exploration in subsequent paragraphs.

The paragraph delves into the specifics of the disk method, explaining how to calculate the volume of a single disk formed by rotating a rectangle around the x-axis. It discusses the transformation of a rectangle into a cylinder and the subsequent calculation of its volume using the area of the circular face and the height of the cylinder. The formula for the volume of a disk is derived, involving the function defining the top surface of the rectangle and the width of the subintervals on the x-axis, leading to an expression involving pi, the square of the function value, and the change in x.

This section discusses the transition from calculating the volume of a single disk to the entire solid formed by adding an infinite number of these disks. It explains how to use an integral to sum the volumes of all disks as the width of each subinterval approaches zero. The process involves setting up a definite integral from the bounds of the region, using the derived formula for a single disk's volume and considering the infinite nature of the sum, which is represented by the integral.

The paragraph presents a practical example of using the disk method to find the volume of a solid obtained by rotating a specific region around the x-axis. The region is bounded by a curve and two vertical lines, and the paragraph guides through determining the interval for the integral, setting up the reference rectangle, and applying the disk method formula to find the volume. It also includes a step-by-step calculation, leading to the final volume of the solid.

This paragraph emphasizes the importance of sketching the solid for a better understanding and then reviews the formula for the disk method when using a vertical slice. It explains the process of reflecting the region across the axis of rotation to visualize the solid and provides a formula for calculating the volume when the region is revolved around the x-axis, using the radius and height obtained from the function and the limits of integration.

The paragraph introduces the washer method, which is used when there's an empty space between the region and the axis of rotation. It explains the concept of a washer, which is like a disk with a hole in the middle, and the strategy of finding the volume of the empty space to then subtract it from the volume of a solid disk. The paragraph sets up the integral for the washer method, considering both the outer and inner radii and their respective volumes.

This section applies the washer method to a problem involving the rotation of a region bounded by two functions around the x-axis. It explains how to find the volume of the outer and inner solids separately and then subtract them to get the volume of the washer. The paragraph includes a step-by-step guide on setting up the integrals for both the outer and inner functions and combining them into a single integral to find the final volume.

The paragraph presents the general formula for the washer method when using a vertical slice and explains the importance of drawing reference rectangles perpendicular to the axis of rotation. It walks through the process of setting up a problem for the washer method, including determining the interval and the expressions for the outer and inner radii, and emphasizes the importance of correctly setting up the integral for accurate volume calculation.

This section discusses the washer method when using a horizontal slice, explaining the difference in setup when the axis of rotation requires horizontal rectangles. It provides a detailed example of setting up and calculating the volume of a solid formed by revolving a region around a horizontal line, including determining the correct expressions for the outer and inner radii and integrating to find the volume.

The final paragraph wraps up the lesson by highlighting the importance of correctly setting up the disk and washer methods, emphasizing the need to draw reference rectangles perpendicular to the axis of rotation. It also poses a critical thinking question about the relationship between the disk and washer methods and the circumstances under which each should be used, encouraging viewers to reflect on the content covered.