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Chapter 6 – Applications of Integration 6.2 Volumes 1Erickson

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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.

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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.

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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.

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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.

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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

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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

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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.

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Find the Volume of the Pyramid 6.2 Volumes9Erickson

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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

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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.

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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

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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

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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

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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

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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

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Volume of a Washer 6.2 Volumes17 where R is the outside radius, r is the inside radius, and h is the height. Erickson

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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

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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

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Examples: Volume by Washer (1) Erickson6.2 Volumes20

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Examples: Volume by Washer (2) Erickson6.2 Volumes21 The outer radius is: R The inner radius is: r

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Examples: Volume by Washer (2) Erickson6.2 Volumes22 R r

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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.

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