6.2 - Volumes Roshan

What is Volume? What do we mean by the volume of a solid? How do we know that the volume of a sphere of radius r is 4πr 3 /3 ? How can we give a precise definition of volume? Roshan

Familiar Cylinders Circular cylinder –Base is a circle of radius r, and so has area πr 2 –Thus V = πr 2 h Rectangular box –Base is a rectangle of area lw –Thus V = lwh The base of a cylinder need not be circular! Roshan

What About Other Solids? What if a solid S is not a cylinder? We can use our knowledge of cylinders: –Cut S into thin parallel slices –Treat each piece as though it were a cylinder, and add the volumes of the pieces –The thinner we make the slices, the closer we will be to the actual volume of S This leads to the method of finding volumes by cross-sections Roshan

The Method of Cross-Sections Intersect S with a plane P x perpendicular to the x-axis Call the cross-sectional area A(x) A(x) will vary as x increases from a to b Divide S into “slabs” of equal width ∆x using planes at x 1, x 2,…, x n Roshan

Cross-Sections The i th slab is roughly a cylinder w/ volume –Here is the base of the “cylinder” andis its height Adding the volumes of the individual slabs gives an approximation to the volume of S: Roshan

Now we use more and more slabs –This corresponds to letting n  ∞ The larger the value of n, the closer each slab becomes to an actual cylinder –In other words, our approximation becomes better and better as n  ∞ So we define the volume of S by Precise Definition of Volume Roshan

Proving the Volume of a Cone Roshan

Example 1 Show the area of a sphere is 4πr 3 /3 Roshan

A Special Case A solid of revolution is formed by rotating a plane region about an axis Roshan

About Solids of Revolution The slabs are always in one of two shapes: –Either the shape of a disk… …as was the case in the sphere problem above –or the shape of a washer A washer is the region between two concentric circles Roshan

Example 2 Find the volume of the solid obtained by rotating about the x-axis the region under the curve from 0 to 1. Roshan

Find the volume of the solid obtained by rotating the region bounded by y = x 3, y = 8, and x = 0 about the y-axis NOTE: slices are perpendicular to the y-axis, rather than the x-axis Example 3 Roshan

Example 4 Find the volume of the solid obtained by rotating about the x-axis the region enclosed by the curves y = x and y = x 2 Roshan

Example 5 Find the volume of the solid obtained by rotating the same region as in the preceding example BUT about the line y = 2 instead of the x-axis Roshan

Example 6 Find the volume of the solid obtained by rotating the region in example 4 about the line x = -1 Roshan

The formula can be applied to any solid for which the cross-sectional area A(x) can be found This includes solids of revolution, as shown above but includes many other solids as well A Bigger Picture Roshan

Example 7a Roshan

Example 7b Roshan Let’s take our same region but this time… The region R is the base of a solid and cross sections perpendicular to the x-axis are –a) semi-circles –b) rectangles with height = 5

A solid has a circular base of radius 1. Parallel cross-sections perpendicular to the base are equilateral triangles. Find the volume of the solid. Example 8 Roshan

Sometimes the disk/washer method is difficult for a solid of revolution An alternative is to divide the solid into concentric circular cylinders This leads to the method of cylindrical shells Method of Cylindrical Shells Roshan

Example of Cylindrical Shells Problem: Suppose the region bounded by y = 2x 2 – x 3 and the x-axis, is rotated about the y-axis To use washers, we must find the inner and outer radii, which would require solving a cubic equation Shells Method: We can find this volume by rotating an approximating rectangle about the y-axis Note that our cross section is now parallel to the axis we are revolving about Roshan