Question

Use the shell method to find the volume of the solid generated by revolving the region bounded by the given curves and lines about the y-axis. Y = 3x, y = -x/3, x = 1 A) 10/9 pi B)20/9 pi C) 10 pi D)19/9 pi Use the shell method to find the volume of the solid generated by revolving the region bounded by the given curves and lines about the x-axis. x = 5 squareroot y, x = -5, y = 1 A) 22/3 pi B) 40/3 pi C) 10 pi D) 11/3 pi Use the shell method to find the volume of the solid generated by revolving the region bounded by the gives curves about the given lines. y =4x, y = 0, x = 2: revolve about the line x = -4 A) 256/3 pi B) 128/3 C) 128/3 pi D) - 128/3 pi

Answer 1

The shell method is a process used in calculus to calculate the volume of a solid of revolution. It involves integrating cylindrical shells and applying the formula V = 2π∫ p(x) q(x) dx. The exact volume is found through setting up proper integrals and using algebraic manipulation.

Step-by-step explanation:

Calculating Volumes Using the Shell Method

The shell method is a technique used to calculate the volume of a solid of revolution by integrating cylindrical shells. The volume V of the solid generated by revolving the region bounded by the curves and lines about the y-axis can be found by the formula V = 2π∫ p(x) q(x) dx, where p(x) is the radius to the y-axis, and q(x) is the height of the shell.

In the first problem, finding the volume of the solid generated by revolving the region bounded by Y = 3x, y = -x/3, and x = 1 around the y-axis involves setting up an integral with the correct limits of integration and functions for p(x) and q(x). After integrating, we will use geometrical knowledge and algebraic manipulation to arrive at the correct answer, which should match one of the provided choices: A) 10/9π, B)20/9π, C) 10π, or D)19/9π.

Similarly, for the second problem revolving around the x-axis with the region defined by x = 5√y, x = -5, and y = 1, and for the third problem revolving around the line x = -4 with the region defined by y = 4x, y = 0, and x = 2, the same process applies using the shell method. Appropriate integrals will be set up, solving for the volume and selecting the correct choice from the options given: A) 22/3π, B) 40/3π, C) 10π, D) 11/3π for the second problem and A) 256/3π, B) 128/3, C) 128/3π, D) - 128/3π for the third problem.