Explanation Given: Consider a cylinder of radius r . Show the expression for the pressure due to volume and expansion of gas inside the cylinder: P = P ( V ) (1) The force exerted by the gas on the piston is shown below: F = Area×Pressure = π r 2 P (2) Calculation: Consider a force function F ( x ) . Express the work done by the force f ( x ) to move an object from a location a to b as shown below: W = ∫ a b f ( x ) d x (3) The gas applies a force F ( x ) on the piston to move it from a location x 1 to x 2 . Calculate the work done to move the piston from x 1 to x 2 . Substitute F ( x ) for f ( x ) , x 1 for a , x 2 for b in Equation (3). W = ∫ x 1 x 2 F ( x ) d x (4) Calculate the volume of the cylinder V as shown below: V = π r 2 x (5) The volume of the cylinder is a function of x . Since pressure depends on volume of the cylinder, it is also a function of x . P = P ( V ( x ) ) (6) Modify Equation (2) using Equation (1) and (6). Substitute π r 2 P for F ( x ) , and P ( V ( x ) ) for P in Equation (2). W = ∫ x 1 x 2 π r 2 P d x W = ∫ x 1 x 2 π r 2 P ( V ( x ) ) d x (7) Differentiate the volume of cylinder with respect to x

Single Variable Calculus: Early Transcendentals, Volume I

8th Edition

ISBN: 9781305270343

Author: James Stewart

Publisher: Cengage Learning

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Chapter 6.4, Problem 29E

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To show: The work done by the gas is W=∫V1V2PdV.

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Chapter 6.4, Problem 28E

Chapter 6.4, Problem 30E

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Single Variable Calculus: Early Transcendentals, Volume I

Ch. 6.1 - Find the area of the shaded region.Ch. 6.1 - Prob. 2E Ch. 6.1 - Prob. 3E Ch. 6.1 - Prob. 4E Ch. 6.1 - Prob. 5E Ch. 6.1 - Prob. 6E Ch. 6.1 - Sketch the region enclosed by the given curves....Ch. 6.1 - Sketch the region enclosed by the given curves....Ch. 6.1 - Sketch the region enclosed by the given curves....Ch. 6.1 - Sketch the region enclosed by the given curves....

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The work done by the gas is W = ∫ V 1 V 2 P d V .

Solution Summary: The author illustrates the work done by the gas on a cylinder of radius r.

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Chapter 6 Solutions