Presentation on theme: "Chapter 6 – Applications of Integration 6.2 Volumes 1Erickson."— Presentation transcript:
Chapter 6 – Applications of Integration 6.2 Volumes 1Erickson
Solids of Revolution Erickson6.2 Volumes2 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.
Volumes of Solids Erickson6.2 Volumes3 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=πr 2 h In other words it is the area of the base times the height.
Volumes of Solids Erickson6.2 Volumes4 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.
Volumes of Solids Erickson6.2 Volumes5 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.
Definition of Volume (Vertical Slices) 6.2 Volumes6 Let S be a solid that lies between x = a and x = b. If the cross-sectional area of S in the plane P x, through x and perpendicular to the x-axis, is A(x), where A is a continuous function, then the volume of S is Erickson
Definition of Volume (Horizontal Slices) 6.2 Volumes7 Let S be a solid that lies between y = c and y = d. If the cross-sectional area of S in the plane P y, through x and perpendicular to the y-axis, is A(y), where A is a continuous function, then the volume of S is Erickson
Volumes of Solids Erickson6.2 Volumes8 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.
Find the Volume of the Pyramid 6.2 Volumes9Erickson
Find the Volume of the Pyramid: 6.2 Volumes Consider a horizontal slice through the pyramid. s dh The volume of the slice is s 2 dh. If we put zero at the top of the pyramid and make down the positive direction, then s=h. 0 h This correlates with the formula where B is the area of the base: Erickson
Method of Slicing Erickson6.2 Volumes11 1 Find a formula for V(x). (Note that I used V(x) instead of A(x).) Sketch the solid and a typical cross section. 2 3 Find the limits of integration. 4 Integrate V(x) to find volume.
Some Useful Areas - Disk 6.2 Volumes12 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 Erickson
Volume of a Disk Erickson6.2 Volumes13 1. 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 2. 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
Example 1: Volume by Disk 6.2 Volumes14 Find the volume of a solid obtained by rotating about the x-axis the region under the curve from 0 to 2. Erickson
Example 2: Volume by Disk 6.2 Volumes15 Find the volume of the solid obtained by rotating the region bounded by the curve and the lines x=0 and x=2. Erickson
Some Useful Areas - Washers Erickson6.2 Volumes16 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
Volume of a Washer 6.2 Volumes17 where R is the outside radius, r is the inside radius, and h is the height. Erickson
Volume of a Washer 6.2 Volumes18 1.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 2.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 Erickson
Examples: Volume by Washer (1) 6.2 Volumes19Erickson The region bounded by and is revolved about the y-axis. Find the volume. Because we are rotating around the y- axis, we need to solve our equations for x. If we use a horizontal slice: The “disk” now has a hole in it, making it a “washer”. The volume of the washer is: outer radius inner radius
Examples: Volume by Washer (1) Erickson6.2 Volumes20
Examples: Volume by Washer (2) Erickson6.2 Volumes21 The outer radius is: R The inner radius is: r
Examples: Volume by Washer (2) Erickson6.2 Volumes22 R r
Examples – pg. 438 Erickson6.2 Volumes23 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.