Finding Volume of Disks (Solids of Revolution) Calcultor

Volume - Disks

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Let S be a solid that extends along x-axis and is bounded on the left and right , respectively, by planes that are perpendicular to the x-axis at x=a and x=b. Find the volume V of the solid, assuming that its cross-sectional area A(x) is known at each point x in the interval [a,b].

To solve this problem, we divide the interval [a,b] into n subintervals, which has the effect of dividing the solid into n slabs

If we assume that the width of the kth slab is

then the volume of the slab can be approximated by the volume of the right cylinder of width (height) delta x of k

and cross sectional area Area of x of k

where

is any point in the kth subinterval. Adding these approximations yields the following Riemann sum that approximates the Volume V Reimann sum that approxiamtes the Volume V

Reimann sum that approxiamtes the Volume V

Taking the limit as n increases and the widths of the subintervals approach zero yields the definite integral

Volume as definite integral

VOLUME FORMULA

Let S be a solid bounded by two parallel planes perpendicular to the x-axis at x=a and x=b. If, for each x in [a,b], the cross-sectional area of S perpendicular to the x-axis is A(x), then the volume of the solid is

Volume Integral Formula

provided A(x) is integrable.

SOLIDS OF REVOLUTION

A solid of revolution is a solid that is generated by revolving a plane region about a line that lies in the same plane as the region , the line is called the axis of revolution. Many familiar solids are of this type

solids of revolution figure 1 solids of revolution figure 2
solids of revolution figure 3 solids of revolution figure 4

Let f be continuous and nonnegatice on [a,b], and let R be the region that is bounded above by y=f(x), below by the x-axis and on the sides by the lines x=a, and x=b. Find the volume of the solid of revolution that is generated by revolving the region R about the x-axis.

Volume example 1a Volume example 1b

We can solve this problem by slicing. For this purpose, observe that the cross section of the solid taken perpendicular to the x-axis at the point x is a circular disk of radius f(x). The area of this region is

A(x) =

Thus the volume of the solid is

Volume

Because the cross sections are disk shaped, the application of this formula is called the method of disks.

EXAMPLE

Find the volume of the solid that is obtained when the region under the curve y= over the interval [1,4] is revolved about the x-axis

SOLUTION

Volume sqrt x x=1 x=4