AI - Volume by Cylindrical Shell Lesson

Volume by Cylindrical Shell

image of cylinder with r1, r2, and h indicated If the rectangular strips that approximate a region revolved about an axis are parallel rather than perpendicular to an axis, a cylindrical shell is created. A technique for computing volume using cross sections parallel to the axis of revolution is the shell method. The presentation below illustrates the rationale behind the using the method of cylindrical shells to find the volume of a solid of revolution.

View the presentations targeting the use of shells to find the volume of a solid of revolution.

The two-part presentation illustrates how the shell method is used when revolving a curve around a line other than one of the axes.

Practical Applications

image of chess pieces Applications of integration are prevalent in physics, manufacturing, economics, biology, and engineering.

In manufacturing many solid objects, especially those made on a lathe, have a circular cross-section and curved sides. A lathe rotates a workpiece on its axis to perform various operations such as cutting, sanding, drilling, or deformation with tools that are applied to the piece of work to create an object that is symmetric about an axis of rotation. Examples of solids of revolution produced in the manufacturing sector include table legs, bowls, chess pieces, baseball bats, pontoons, cue sticks, woks, and candlestick holders.

Consumer surplus and producer surplus are frequent applications of integration in economics. Consumer surplus is the difference between the price consumers are willing to pay for particular goods or services and the actual price. Producer surplus is the difference between the amount for which producers are willing and able to supply a good and the price they actually receive. Consumer surplus and producer surplus are calculated using the supply and demand curves. Assuming that the equilibrium price is P₀ and the equilibrium quantity is Q₀, the formulas for consumer surplus and producer surplus are

equation image indicator $LaTeX: CS=\int_0^{Q_0}D\left(Q\right)-P_0\left(Q_0\right)\\ PS=P_0\left(Q_0\right)-\int_0^{Q_0}S\left(Q\right)$ $CS=\int_0^{Q_0}D\left(Q\right)-P_0\left(Q_0\right)\\ PS=P_0\left(Q_0\right)-\int_0^{Q_0}S\left(Q\right)$

where D(Q) represents the demand function written in terms of Q and S(Q) represents the supply function written in terms of Q. View the presentation on consumer and producer surplus.

Another application of integration relates to the human cardiovascular system. The cardiac output of the heart is the volume of blood pumped by the heart per unit time, i.e., the rate of flow into the aorta. When an indicator (dye) is injected into the right atrium of the heart, it flows through the heart into the aorta. A probe is inserted into the aorta to measure the concentration of dye leaving the heart at equally spaced times over a time interval [0, T] until the dye has cleared. The output of the heart is equal to the amount of indicator (dye) injected divided by its average concentration in the arterial blood after a single circulation through the heart. More specifically, cardiac output is equal to the quantity of indicator dye injected divided by the area under the dilution curve measured downstream from the injection site.

equation image indicator $LaTeX: \text{Cardiac Output}=\frac{\text{Quantity of indicator}}{\int_0^T\text{(Concentration of indicator)}dt}$ $\text{Cardiac Output}=\frac{\text{Quantity of indicator}}{\int_0^T\text{(Concentration of indicator)}dt}$

Volume by Cylindrical Shell Practice

Volume by Cylindrical Shell: Even More Problems!

Complete problems from your textbook and/or online resources as needed to ensure your complete understanding of volumes by cylindrical shells.

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