If, however, we rotate the region around a line other than the y -axis, y -axis, we have a different outer and inner radius. This was based on a shell with an outer radius of x i x i and an inner radius of x i − 1. V shell = f ( x i * ) ( π x i 2 − π x i − 1 2 ) = π f ( x i * ) ( x i 2 − x i − 1 2 ) = π f ( x i * ) ( x i + x i − 1 ) ( x i − x i − 1 ) = 2 π f ( x i * ) ( x i + x i − 1 2 ) ( x i − x i − 1 ). Previously, regions defined in terms of functions of x x were revolved around the x -axis x -axis or a line parallel to it. Note that this is different from what we have done before. We then revolve this region around the y-axis, as shown in Figure 6.25(b). As before, we define a region R, R, bounded above by the graph of a function y = f ( x ), y = f ( x ), below by the x -axis, x -axis, and on the left and right by the lines x = a x = a and x = b, x = b, respectively, as shown in Figure 6.25(a). The Method of Cylindrical ShellsĪgain, we are working with a solid of revolution. In the last part of this section, we review all the methods for finding volume that we have studied and lay out some guidelines to help you determine which method to use in a given situation. Also, the specific geometry of the solid sometimes makes the method of using cylindrical shells more appealing than using the washer method. The ability to choose which variable of integration we want to use can be a significant advantage with more complicated functions. With the method of cylindrical shells, we integrate along the coordinate axis perpendicular to the axis of revolution. We can use this method on the same kinds of solids as the disk method or the washer method however, with the disk and washer methods, we integrate along the coordinate axis parallel to the axis of revolution. In this section, we examine the method of cylindrical shells, the final method for finding the volume of a solid of revolution. 6.3.2 Compare the different methods for calculating a volume of revolution.6.3.1 Calculate the volume of a solid of revolution by using the method of cylindrical shells.

Given height and volume calculate the radius, lateral surface area and total surface area.

Given height and lateral surface area calculate the radius, volume and total surface area.

Given radius and lateral surface area calculate the height, volume and total surface area.

Given radius and volume calculate the height, lateral surface area and total surface area.

Given radius and height calculate the volume, lateral surface area and total surface area.

Use the following additional formulas along with the formulas above. You can do this using theĬircle calculator. To calculate the total surface area you will need to also calculate the area of the top and bottom. ** The area calculated is only the lateral surface of the outer cylinder wall.

Total surface area of a closed cylinder is:.

Calculate the top and bottom surface area of a cylinder (2.

Calculate the lateral surface area of a cylinder (just the curved outside)**:.

Calculations are based on algebraic manipulation of these standard formulas. For example, if you are starting with mm and you know r and h in mm, your calculations will result with V in mm 3, L in mm 2, T in mm 2, B in mm 2 and A in mm 2.īelow are the standard formulas for a cylinder. The units are in place to give an indication of the order of the results such as ft, ft 2 or ft 3.

Units: Note that units are shown for convenience but do not affect the calculations. This is a right circular cylinder where the top and bottom surfaces are parallel but it is commonly referred to as a 'cylinder.' It will also calculate those properties in terms of PI π. This online calculator will calculate the various properties of a cylinder given 2 known values.

Calculating volume of cylindrical tank