Chapter 6 – Applications of Integration 6.2 Volumes 6.2 Volumes Erickson

Solids of Revolution Solids generated by revolving plane regions around the axes are called solids of revolution. Examples: billiard balls, threaded spools etc. We can find their volume by using geometry but here we are going to learn how to use calculus to find the volume. 6.2 Volumes Erickson

Volumes of Solids In this section we will learn how to find volumes of solids by using integration. Here again we will work with areas. Think for a second, how do we find the volume of a solid? Ex. the volume of a cylinder is: V=πr2h In other words it is the area of the base times the height. 6.2 Volumes Erickson

where A is the area of the base. Volumes of Solids The same is true for other cylinder solids: Examples In each case the volume would be V=Ah where A is the area of the base. 6.2 Volumes Erickson

Volumes of Solids If we can set things up so that the axis of revolution is the x- axis and the region is the region of the plane between the x-axis and the graph of a continuous function y = A(x) a ≤ x ≤ b, we can calculate the volume of the solids by approximation. We can partition the solid in n vertical rectangles and find the area of each. The sum of those areas will give us an approximation of the volume. 6.2 Volumes Erickson

Definition of Volume (Vertical Slices) Let S be a solid that lies between x = a and x = b. If the cross-sectional area of S in the plane Px, through x and perpendicular to the x-axis, is A(x), where A is a continuous function, then the volume of S is 6.2 Volumes Erickson

Definition of Volume (Horizontal Slices) Let S be a solid that lies between y = c and y = d. If the cross-sectional area of S in the plane Py, through x and perpendicular to the y-axis, is A(y), where A is a continuous function, then the volume of S is 6.2 Volumes Erickson

Volumes of Solids When we use the volume formula, it is important to remember that A(x) is the area of a moving cross-section obtained by slicing through x perpendicular to the x-axis. Similarly, when we use the volume formula, it is important to remember that A(y) is the area of a moving cross-section obtained by slicing through y perpendicular to the y-axis. 6.2 Volumes Erickson

Find the Volume of the Pyramid 6.2 Volumes Erickson

Find the Volume of the Pyramid: 4 Consider a horizontal slice through the pyramid. The volume of the slice is s2dh. If we put zero at the top of the pyramid and make down the positive direction, then s=h. h This correlates with the formula where B is the area of the base: s dh 6.2 Volumes Erickson

Method of Slicing 1 Sketch the solid and a typical cross section. Find a formula for V(x). (Note that I used V(x) instead of A(x).) 2 3 Find the limits of integration. 4 Integrate V(x) to find volume. 6.2 Volumes Erickson

Some Useful Areas - Disk If the cross-section is a disk, we find the radius of the disk (in terms of x or y) and use A = π (radius)2 6.2 Volumes Erickson

Volume of a Disk If the solid consists of adjacent vertical disks between x = a and x = b, we find the radius R(x) of the disk at x, and the volume is If the solid consists of adjacent horizontal disks between y = c and x = d, we find the radius R(y) of the disk at y, and the volume is 6.2 Volumes Erickson

Example 1: Volume by Disk Find the volume of a solid obtained by rotating about the x-axis the region under the curve from 0 to 2. 6.2 Volumes Erickson

Example 2: Volume by Disk Find the volume of the solid obtained by rotating the region bounded by the curve and the lines x=0 and x=2. 6.2 Volumes Erickson

Some Useful Areas - Washers If the cross-section is a washer, we find the inner radius and the outer radius of the washer (in terms of x or y) and use A = π (outer radius)2 − π (inner radius)2 6.2 Volumes Erickson

Volume of a Washer where R is the outside radius, r is the inside radius, and h is the height. 6.2 Volumes Erickson

Volume of a Washer If the solid consists of adjacent vertical washers between x = a and x = b, we find the outside radius R(x) and inside radius r(x) of the washer at x, and the volume is If the solid consists of adjacent horizontal washers between y = c and x = d, we find the outside radius R(y) and inside radius r(y) of the disk at y, and the volume is 6.2 Volumes Erickson

Examples: Volume by Washer (1) The region bounded by and is revolved about the y-axis. Find the volume. If we use a horizontal slice: The “disk” now has a hole in it, making it a “washer”. Because we are rotating around the y-axis, we need to solve our equations for x. The volume of the washer is: outer radius inner radius 6.2 Volumes Erickson

Examples: Volume by Washer (1) 6.2 Volumes Erickson

Examples: Volume by Washer (2) The outer radius is: r The inner radius is: R 6.2 Volumes Erickson

Examples: Volume by Washer (2) 6.2 Volumes Erickson

Examples – pg. 438 Find the volume of the solid obtained by rotating the region bounded by the given curves about the specified line. Sketch the region, the solid, and a typical disk or washer. 6.2 Volumes Erickson