10.7.1 Introducing the Shell Method

Question 1

Introducing the Shell Method

Answer 1

• Using the shell method, the volume V of a solid of revolution is given by , where x is the radius and
  h(x) is the height of an arbitrary shell.

Question 2

note

Answer 2

• You have already found the volume of this solid of revolution using the washer method. But is there another way to determine the volume?
• Instead of slicing the volume into washers, what happens if you core the solid instead? Think of cutting little cylinders out of the solid.
• Notice that the solid can be described by an infinite number of cylinders whose radius and height are given by the equations bounding the rotated region.
• The method of finding volume using these cylinders is called the shell method.
• Consider the volume of a thin right cylindrical shell. The shell itself can be cut open. When you do this, you are left with a rectangular solid with a very tiny thickness.
• The volume of a shell is equal to the circumference of the circle times the height times the thickness. The thickness is a small change in x.
• The circumference is equal to 2 r, where r is equal to the distance from the axis of revolution. Here, r = x.
• The height of the cylinder is equal to the distance between the parabola and the x-axis. This distance is given by x 2 – 4.
• Multiplying all those pieces together gives you the volume of an arbitrary cylindrical shell.
• To find the volume, just integrate. Notice that the limits of integration run along the x-axis, since that is where the shells stack.

Question 3

Use the method of cylindrical shells to find the volume of the solid of revolution generated by rotating the region bounded by the curves y=sin(x2) and y=−sin(x2) for0≤x≤√π about the y=axis.

Answer 3

4π

Question 4

For b > a, which of the following equations representing a two-dimensional curve in the xy-plane would generate a torus when rotated about the given axis of rotation?

Answer 4

(x − b)^2 + y ^2 ≤ a ^2 rotated around x = 0

Question 5

Determine the volume of the solid of revolution generated by revolving the ellipse x^2/a^2+y^2/b^2=1, where a>b,around the x-axis using the method of cylindrical shells.

Answer 5

4/3πab^2

Question 6

Consider the two functions y=f(x)and y=g(x), where f(x)>0, g(x)>0,and f(x)>g(x) for x∈[a,b] as shownin the figure .Which of the following correctly formulates the shell method to calculatethe volume of the solid revolution generated by rotating the region bounded by the given functions aboutx=0?

Answer 6

2π∫^b_a x[f(x)−g(x)]dx

Question 7

Given the function y = f (x), which can be expressed as x = g ( y), where f (x) > 0 and g ( y) > 0, which of the following correctly formulates the shell method to calculate the volume of the solid of revolution generated by rotating the region bounded by the given curve about the y‑axis?

Answer 7

2π∫bax⋅f(x)dx

Question 8

Calculate the volume of the solid of revolution generated by revolving the region bounded by the parabolas y 2 = 2 (x − 3) and y 2 = x about y = 0

Answer 8

9π