Let R and H be positive numbers. In questions (1) and (2), the region R is the triangular region determined by the vertices (0,0), (0, R) and (H, R) in the xy-plane. (1) Use the Washer Method to find the volume of the solid obtained when R is rotated about the x axis. (2) Use the Shell Method to find the volume of the solid obtained when R is rotated about the 3 axis. Is there a way to check the accuracy of the answers to questions (1) and (2) using the formula for the volume of a cone?

V = 2*pi*R^2*H / 3   .... Accurate according to formula

Step-by-step explanation:

Given:

- Coordinates of the triangular region:

                             ( 0 , 0 ) , ( 0 , R ) , ( H , R )

- The region is rotated about x-axis.

Find:

Use the Washer Method to find the volume of the solid obtained when R is rotated about the x axis.

Use the Shell Method to find the volume of the solid obtained when R is rotated about the x axis

Is there a way to check the accuracy of the answers to questions (1) and (2) using the formula for the volume of a cone?

Solution:

- Using washer method. Define functions f(x) and g(x), such that f(x) > g(x):

                                  f(x) = R

                                  g(x) = ( R / H )*x

- Set up the integral for washer method as follows:

- Using the shell method we have to express the area A as a function of y first:

                                 A(y) = 2*pi*Radius*Width

                                 A(y) = 2*pi*R*H - 2*pi*y*(H-H*y/R)

                                 A(y) = 2*pi*R*H - 2*pi*(H*y-H*y^2/R)

- Set up the integral and evaluate:

- The volume of the solid can be verified by removing a cone from a cylinder as follows:

                                 V = V_cylinder - V_cone

                                 V = pi*R^2*H - pi*R^2*H / 3

                                 V = 2*pi*R^2*H / 3

- The volume of solid derived is accurate according to the formulas.