revolution mc-TY-volumes-2009-1 We sometimes need to calculate the volume of a solid which can be obtained by rotating a curve about the x-axis There is a
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[PDF] Volumes of solids of revolution - Mathcentre
revolution mc-TY-volumes-2009-1 We sometimes need to calculate the volume of a solid which can be obtained by rotating a curve about the x-axis There is a
[PDF] Volumes by Integration
The 3-D model of the solid of revolution FORMULAS: V= ∫Adx , or respectively ∫Ady where A stands for the area of the typical disc and r=f(x) or r=f(y) depending on the axis of revolution
[PDF] Volumes of Solids of Revolution - York University
We then rotate this curve about a given axis to get the surface of the solid of revolution Lets rotate the curve about the x-axis We want to determine the volume of
[PDF] 76 Finding the Volume of a Solid of Revolution—Disks Introduction
Revolve the graph about the x-axis Let's investigate a typical infinitesimal slice of the resulting solid of revolution the slice is a disk with volume π(
[PDF] 72 Volumes of Revolution Using the Disk Method How could we
3 Volume of a Solid formed by rotation of a region around a horizontal or vertical line How could we find the volume of the solid created by revolving this curve
[PDF] L37 Volume of Solid of Revolution I Disk/Washer and Shell Methods
Two common methods for finding the volume of a solid of revolution are the ( cross sectional) disk method and the (layers) of shell method of integration To apply
[PDF] Volume of Solids of Revolution from section 133
Volume of Solids of Revolution from section 13 3 Consider a region R in the xy- plane Take any point (x,y) of the region If we rotate this point about the x-axis , it
[PDF] Area Between Curves Volumes of Solids of Revolution
Volumes of Solids of Revolution Area Between Curves Theorem: Let f(x) and g(x ) be continuous functions on the interval [a, b] such that f(x) ≥ g(x) for all x
[PDF] Calculus Online Textbook Chapter 8 - MIT OpenCourseWare
Cones and spheres and circular cylinders are "solids of revolution volume of solid of revolution = ry2 dx J = solid volume = integral of shell volumes = (4)
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Volumes of solids ofrevolution
mc-TY-volumes-2009-1 We sometimes need to calculate the volume of a solid which canbe obtained by rotating a curve about thex-axis. There is a straightforward technique which enables this to be done, using integration. In order to master the techniques explained here it is vital that you undertake plenty of practice exercises so that they become second nature. After reading this text, and/or viewing the video tutorial on this topic, you should be able to: find the volume of a solid of revolution obtained from a simplefunctiony=f(x)between given limitsx=aandx=b; find the volume of a solid of revolution obtained from a simplefunctiony=f(x)where the limits are obtained from the geometry of the solid.Contents
1.Introduction 2
2.The volume of a sphere 4
3.The volume of a cone 4
4.Another example 5
5.Rotating a curve about they-axis 6
www.mathcentre.ac.uk 1c?mathcentre 20091. IntroductionSuppose we have a curve,y=f(x).
y = f(x) x = a x = b Imagine that the part of the curve between the ordinatesx=aandx=bis rotated about the x-axis through360◦. The curve would then map out the surface of a solid as it rotated. Such solids are calledsolids of revolution. Thus if the curve was a circle, we would obtain the surface of a sphere. If the curve was a straight line through the origin, we would obtain the surface of a cone. Now we already know what the formulae for the volumes of a sphere and a cone are, but where did they come from? How can they calculated? If we could find a general method for calculating the volumes of the solids of revolution thenwe would be able to calculate, for example, the volume of a sphere and the volume of a cone, as well as the volumes of more complex solids. To see how to carry out these calculations we look first at the curve, together with the solid it maps out when rotated through360◦. y = f(x) Now if we take a cross-section of the solid, parallel to they-axis, this cross-section will be acircle. But rather than take a cross-section, let us take a thin disc of thicknessδx, with the face
of the disc nearest they-axis at a distancexfrom the origin. www.mathcentre.ac.uk 2c?mathcentre 2009 y = f(x) x = a x = b δx xy y + δyThe radius of this circular face will then bey. The radius of the other circular face will bey+δy,
whereδyis the change inycaused by the small positive increase inx,δx. The disc is not acylinder, but it is very close to one. It will become even closer to one asδx, and henceδy, tends
to zero. Thus we approximate the disc with a cylinder of thickness, or height,δx, and radiusy. The volumeδVof the disc is then given by the volume of a cylinder,πr2h, so that