In analytical geometry, the formula to calculate the volume of a circular cylinder, rectangular box and sphere were found to be:

• Circular cylinder: V = πr²h
• Rectangular box: V = wdh
• Sphere: V = 4/3πr³    

In this section, the formula to calculate the volume of a solid of revolution and a cylindrical shell are introduced.

The Cross-section Area A(x)

Definition of Volume

Before the volume of a sold of revolution can be calculated, some terms and regions need to be defined. The intersection of any body S with a plane P is a plane region called a cross-section of S. If P is perpendicular to the x-axis and passes through the point x on the x-axis, the area of the cross-section region is A(x), where x on the interval [a, b]. The area A(x) will very as x increases from a to b.

Generally, it is easier to find the volume of a small uniform region and then sum them together to get the total volume. First, divide the interval [a, b] into n subintervals.

      Δx = (b - a)/n

In each subinterval, choose a point x_i^*. On each subinterval [x_i, x_i+1], the volume can be approximated as a volume of a disk which has a face area of A(x_i^*) and height Δx.

      V_i = A(x_i^*)Δx

Then the volume of the region where x is on interval [a, b] can be approximated as

If A(x) is continuous, the limit of the above Rieman sum exits. The exact volume is given by

In a similar way, a cross-section area can be generated about y-axis using A(y) where y is in an interval [c, d]. The volume is given by

Solids of Revolution

A Circular Disk Cross-section Perpendicular to x-axis

A solid of revolution is obtained by revolving a region in a plane about a line that does not intersect the region. The line about which the rotation takes place is called the axis of revolution. Considering a curve defined by a function, y = f(x), where x is on an interval [a, b]. Rotate the region bounded by the curves y = f(x), y = 0, x = a, and x = b about x-axis. Then a solid body S is generated. Because S is obtained by rotation, a cross-section through x on the x-axis perpendicular to the x-axis is a circular disk with radius

and the cross-sectional area of the circular disk is

      A(x) = πy² = π[f(x)]²

Using the basic volume formula derived above, the formula for a volume of revolution is

In a similar way, if rotate a curve defined by a function x = g(y), where y is on an interval [c,d], the volume of this revolution is

Volume of Revolution by y = f(x) - g(x)

If rotating a region bounded by the curves y = f(x), y = g(x), x = a, and x = b about the x-axis, a solid body with a "hole" in the middle S is generated. The volume of this body can be obtained by subtracting volumes shown below.

If V₁ is the volume of the revolution by rotating the curve y = f(x), where x is on an interval [a, b] and V₂ is the corresponding volume for the curve y = g(x), where x is on an interval [a, b], the volume of S is calculated by subtracting V₂ from V₁

Cylindrical Shell

Volume by Cylindrical Shell

A Cylindrical Shell Cross-section Perpendicular to y-axis

Some volume problems are difficult to handle using the method of revolution. Another method called cylindrical shells can be easier in these cases.

First consider a cylindrical shell with inner radius r₁ and outer radius r₂, and height h. The volume V is calculated by subtracting the volume of the inner cylinder from the volume of the outer cylinder.

      V = (π r₂²h - π r₁²h) = 2π(r₂ + r₁)/2h(r₂ - r₁) = 2πrhΔr

where r = (r₂ + r₁)/2 is the average radius of the shell, Δr = r₂ - r₁ is the thickness of the shell, and 2πr is the average circumference of the face of the shell.

Then consider a solid body obtained by rotating about the y-axis the region bounded by y = f(x) 0, y = 0, x = a, and x = b (b > a 0). Divide the interval [a, b] to n subintervals,

      Δx = (b - a)/n

Choose a value, x_i^* in the middle of each subinterval, then the volume corresponding to f(x) between [x_i, x_i+1] can be approximated as a cylindrical shell with a height of f(x_i^*), a thickness of Δx, and an average radius x_i^* = (x_i + x_i+1)/2.

      V_i = 2πx_i^* f(x_i^*)Δx

An approximation to the volume V of S is given by the sum of the volumes of these shells

The exact volume is obtained when n approach infinity in the above expression.