Use the shell method to set up and evaluate the integral that gives the volume of the solid generated by revolving...

We can use a shell method when a bounded region represented by rectangular strip is parallel to the axis of revolution. It forms of infinite number of thin hollow pipes or “representative cylinders”.

 In this method, we follow the formula: (length * height * thickness)

or radius*height*thickness

For the bounded region, as shown on the attached image, the rectangular strip is parallel to x-axis (axis of rotation). We...

We can use a shell method when a bounded region represented by rectangular strip is parallel to the axis of revolution. It forms of infinite number of thin hollow pipes or “representative cylinders”.

 In this method, we follow the formula: (length * height * thickness)

or radius*height*thickness

For the bounded region, as shown on the attached image, the rectangular strip is parallel to x-axis (axis of rotation). We can let:

or

The will be based from the boundary line x=0.

The will be base on the equation rearranged into

For boundary values, we have to (based from the boundary line).

Plug-in the values on

  *radius*height*thickness, , we get:

Apply basic integration property:

Apply Law of Exponent: then and y^n*y^m = y^(n+m)

Apply power rule for integration:

Apply definite integration formula: int_a^b f(y) dy= F(b)-F(a).

 or (approximated value).