Representative Rectangles Compare the representative rectangles for the disk and shell methods.

Name: Problem 2, Chapter 7: Calculus of a Single Variable
Uploaded: 2019-07-02T16:50:18-08:00
Duration: 3 min 34 s
Channel: Sarah Davis
Description: Representative Rectangles Compare the representative rectangles for the disk and shell methods.

Representative Rectangles Compare the representative...

Key Concepts

Representative Rectangles

These are small, abstract segments of a region that are used to approximate and eventually exactly calculate areas or volumes as the limit of sums. In integration, they form the basic units whose properties (like area or volume) are summed to approximate a continuous object.

Disk Method

This is an integration technique used to calculate the volume of a solid of revolution. It involves slicing the region perpendicular to the axis of rotation, creating circular disks (or washers if there is a hole). The volume of each disk is computed using the area of the circular cross-section multiplied by its thickness, and integration sums these contributions over the entire region.

Shell Method

This integration method calculates the volume of a solid of revolution by slicing the region parallel to the axis of rotation, forming cylindrical shells. Each shell’s volume is determined by its lateral surface area multiplied by its thickness. The method is particularly useful when the representative slices (rectangles) naturally adapt to cylindrical coordinates, providing a simpler integration setup in some scenarios.

Comparison of Representative Rectangles in Disk and Shell Methods

In the disk method, the representative rectangle is oriented perpendicular to the axis of rotation, forming the base of the disk whose area is calculated. In contrast, the shell method uses rectangles that are parallel to the axis, forming the height of the cylindrical shells. The key difference lies in the orientation and resulting geometry of the slices, which in turn affects the setup of the integral for volume calculation.

Transcript

00:01 Okay, so this is from section 7 .3, problem 2, and it's just a discussion of the difference between the representative rectangles and the disk and shell methods to find volumes of surfaces or solids of revolution.

00:16 So on the left -hand side here, this is a picture of how the disk method works.

00:21 So on the far left, if we rotate around the y -axis, our disks are represented by horizontal.

00:31 Rectangles and if there's a gap that's the smaller radius and then the rectangle itself include and then from the axis of rotation all the way to the edge of the representative rectangle is the larger radius and then this is just using the formula pi r squared to find the area of circles and similarly if we rotate around the x -axis or another horizontal axis the formula just uses vertical representative rectangles in this case and we integrate with respect to x whereas when we rotate around a vertical axis you have to integrate with respect to y and a good thing to notice here is with the disk method the rectangle is perpendicular to the axis of rotation so if you're in a problem where you want to integrate with with respect to y, rotating around a vertical, and you're rotating around a vertical axis, the disk method is great.

01:54 If you're rotating around a horizontal axis, the disk method makes sure that you integrate with respect to x.

02:01 Now on the other hand, in the shell method, the representative rectangles are actually have a parallel shape to the axis of rotation.

02:12 You have vertical axis of rotation, the rectangle is more vertical.

02:16 If you have a horizontal axis of rotation, of rotation, the rectangle is more horizontal.

02:21 And when we rotate around a vertical axis in the shell method, we have to integrate with respect to x...

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