14.3. Volumes of Revolution. Introduction. Prerequisites. Learning Outcomes

Transcription

1 Volumes of Revolution 14.3 Introduction In this Section we show how the concept of integration as the limit of a sum, introduced in Section 14.1, can be used to find volumes of solids formed when curves are rotated around the or ais. Prerequisites Before starting this Section ou should... Learning Outcomes On completion ou should be able to... be able to calculate definite integrals understand integration as the limit of a sum calculate volumes of revolution 2 HELM (28):

2 1. Volumes generated b rotating curves about the -ais Figure 8 shows a graph of the function = 2 for between and 3. 6 = 2 O 3 Figure 8: A graph of the function = 2, for 3 Imagine rotating the line = 2 b one complete revolution (36 or 2π radians) around the -ais. The surface so formed is the surface of a cone as shown in Figure 9. Such a three-dimensional shape is known as a solid of revolution. We now discuss how to obtain the volumes of such solids of revolution. 6 = 2 O 3 Figure 9: When the line = 2 is rotated around the ais, a solid is generated Find the volume of the cone generated b rotating = 2, for 3, around the -ais, as shown in Figure 9. In order to find the volume of this solid we assume that it is composed of lots of thin circular discs all aligned perpendicular to the -ais, such as that shown in the diagram. From the diagram below we note that a tpical disc has radius, which in this case equals 2, and thickness δ. HELM (28): Section 14.3: Volumes of Revolution 21

3 6 = 2 (, ) δ O 3 The cone is divided into a number of thin circular discs. The volume of a circular disc is the circular area multiplied b the thickness. Write down an epression for the volume of this tpical disc: π(2) 2 δ = 4π 2 δ To find the total volume we must sum the contributions from all discs and find the limit of this sum as the number of discs tends to infinit and δ tends to zero. That is =3 lim δ = 4π 2 δ This is the definition of a definite integral. Write down the corresponding integral: 3 4π 2 d Find the required volume b performing the integration: 22 HELM (28):

4 [ 4π 3 3 ] 3 = 36π A graph of the function = 2 for between and 4 is shown in the diagram. The graph is rotated around the -ais to produce the solid shown. Find its volume. 16 = 2 (, ) O δ 4 The solid of revolution is divided into a number of thin circular discs. As in the previous, the solid is considered to be composed of lots of circular discs of radius, (which in this eample is equal to 2 ), and thickness δ. Write down the volume of each disc: π( 2 ) 2 δ = π 4 δ Write down the epression which represents summing the volumes of all such discs: =4 π 4 δ = Write down the integral which results from taking the limit of the sum as δ : HELM (28): Section 14.3: Volumes of Revolution 23

5 4 π 4 d Perform the integration to find the volume of the solid: 4 5 π 5 = 24.8π In general, suppose the graph of () between = a and = b is rotated about the -ais, and the solid so formed is considered to be composed of lots of circular discs of thickness δ. Write down an epression for the radius of a tpical disc: Write down an epression for the volume of a tpical disc: π 2 δ The total volume is found b summing these individual volumes and taking the limit as δ tends to zero: =b lim δ =a π 2 δ Write down the definite integral which this sum defines: b a π 2 d 24 HELM (28):

6 Ke Point 5 If the graph of (), between = a and = b, is rotated about the -ais the volume of the solid formed is b a π 2 d Eercises 1. Find the volume of the solid formed when that part of the curve between = 2 between = 1 and = 2 is rotated about the -ais. 2. The parabola 2 = 4 for 1, is rotated around the -ais. Find the volume of the solid formed. s 1. 31π/5, 2. 2π. 2. Volumes generated b rotating curves about the -ais We can obtain a different solid of revolution b rotating a curve around the -ais instead of around the -ais. See Figure 1. () δ (, ) O Figure 1: A solid generated b rotation around the -ais To find the volume of this solid it is divided into a number of circular discs as before, but this time the discs are horizontal. The radius of a tpical disc is and its thickness is δ. The volume of the disc will be π 2 δ. The total volume is found b summing these individual volumes and taking the limit as δ. If the lower and upper limits on are c and d, we obtain for the volume: =d lim δ =c π 2 δ which is the definite integral d c π 2 d HELM (28): Section 14.3: Volumes of Revolution 25

7 Ke Point 6 If the graph of (), between = c and = d, is rotated about the -ais the volume of the solid formed is d c π 2 d Find the volume generated when the graph of = 2 between = and = 1 is rotated around the -ais. Using Ke Point 6 write down the required integral: 1 π 2 d This integral can be written entirel in terms of, using the fact that = 2 to eliminate. Do this now, and then evaluate the integral: 1 π 2 d = 1 [ π 2 π d = 2 ] 1 = π 2 Eercises 1. The curve = 2 for 1 < < 2 is rotated about the -ais. Find the volume of the solid formed. 2. The line = 2 2 for 2 is rotated around the -ais. Find the volume of revolution. s 1. 15π π HELM (28):