[PDF] L37 Volume of Solid of Revolution I Disk/Washer and Shell Methods

Volumes of solids of revolution

The volume δV of the disc is then given by the volume of a cylinder, πr2h, so that δV = πy2δx So the volume V of the solid of revolution is given by V = lim δx→0 Xx=b x=a δV = lim δx→0 Xx=b x=a πy2δx = Z b a πy2dx, where we have changed the limit of a sum into a definite integral, using our definition of inte-gration

L37 Volume of Solid of Revolution I Disk/Washer and Shell Methods

L37 Volume of Solid of Revolution I Disk/Washer and Shell Methods A solid of revolution is a solid swept out by rotating a plane area around some straight line (the axis of revolution) Two common methods for nding the volume of a solid of revolution are the (cross sectional) disk method and the (layers) of shell method of integration

Infinite Calculus - Finding Volumes of Solids of Revolution

Apr 04, 2015 · Finding Volumes of Solids of Revolution Name_____ ©O B2W0P1z5R TKButt[ai ZSjoxf\tewUaPrmeR fLsLYCb J Z CAilElD orDitg`hXtqsO WrZeAsAetrOvmeAd\ -1-For each problem, find the volume of the solid that results when the region enclosed by the curves is revolved about the the x-axis 1) y = 2x + 3, y = 0, x = 0, x = 2 x y

Practice Problems on Volumes of Solids of Revolution

d Use the Cylindrical Shell Method to find the volume of the solid obtained by rotating the region bounded by y = 1 - x 2, the x-axis, and the y-axis in the first quadrant rotated about the y-axis V = 2 ∫ 1 0 x 1 - x 2 dx = - 2 3 (1 - x ) 3/2 1 0 = 2 /3 e Bounded by y = e 2x and y = e-2x on [0, 2] rotated about the x-axis

Volume of Solids - University of Utah

30B Volume Solids 8 EX 4 Find the volume of the solid generated by revolving about the line y = 2 the region in the first quadrant bounded by these parabolas and the y-axis (Hint: Always measure radius from the axis of revolution )

Calculus 221 worksheet Volume of Solid of revolution

Calculus 221 worksheet Volume of Solid of revolution You might skip the examples with shell method, depending on whether we will go over that in lecture Example 1 The area underneath y= p R2 x2 is revolved around the x-axis Find the volume of the resulting solid, and identify the solid Solution: The curve y= p R2 x2 is de ned from x= Rto x

Volume:The Disk Method - KSU

Solid of revolution with hole R(x) r(x) Plane region ab V b a R x 2 r x 2 dx R x r x, Volume of washer 2R r2 w w r R 7 2 Volume: The Disk Method 449 Axis of revolution R r w r R Disk Solid of revolution w Figure 7 18 y = x2 y = x r = x2 R = x x 1 1 Δx (0, 0) (1, 1) Plane region y −1 1 Solid of revolution x y Solid of revolution Figure 7 20

Volumes of Revolution Q1, (OCR 4723, Jun 2006, Q9)

(a) Find the exact volume of the solid of revolution formed by rotating R completely about the x-axis (b) The region R is rotated completely about the y-axis Explain why the volume of the solid of revolution formed is given by — It e2y dy, and find this volume (i) Find (ii) (x + cos 2x)2 dr

Volumes by Integration - RIT

1 Finding volume of a solid of revolution using a disc method 2 Finding volume of a solid of revolution using a washer method 3 Finding volume of a solid of revolution using a shell method If a region in the plane is revolved about a given line, the resulting solid is a solid of revolution, and the line is called the axis of revolution

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L37 Volume of Solid of Revolution I

Disk/Washer and Shell Methods

Asolid of revolutionis a solid swept out by

rotating a plane area around some straight line (the axis of revolution).

Two common methods for nding the volume of a

solidofrevolutionarethe(crosssectional)disk method and the (layers) ofshell methodof integration.

To apply these methods, it is easiest to:

1. Draw the plane region in question;

2. Identify the area that is to be revolved about the

axis of revolution;

3. Determine the volume of either a disk-shaped slice

or a cylindrical shell of the solid;

4. Sum up the innitely many disks or shells.

V=Z dV 1

Disk method

The volumeVof the solid formed by rotating a plane area about thexaxis is given byV=Z b a

A(x)dx=Z

b a f2(x)dxand about theyaxis byV=Z b a

A(y)dy=Z

b a g2(y)dywhereA(x) andA(y) is the cross-sectional area of the solid. 2 ex.Find the volume of the solid generated when the area bounded by the curvey=px, thexaxis and the linex= 2 is revolved about thexaxis.(2unit3)3

Washer Method

Alternatively, the volume of the solid formed by

rotating the area between the curves off(x) (on top) andg(x) (on the bottom) and the linesx=aand x=babout thexaxis is given by V=Z b a [f2(x)g2(x)]dxThat is, we use 'washers' instead of 'disks' to obtain the volume of the 'hollowed' solid by taking the volume of the inner solid and subtract it from the volume of the outer solid. 4 Note:

1.f2g26= (fg)2

2. To rotate about any horizontal axis, we must rst

calculate the outer radius (OR) and the inner ra-

dius (IR), then use the area of a washerA=[(O:R:)2(I:R)2]to give us the volume of the solid of revolution

V=Z b a [(O:R:)2(I:R)2]dxO.R.(Outer Radius) = Distance from the axis of revolution to the outer edge of the solid;

I.R.(Inner Radius) = Distance from the axis of

revolution to the inner edge of the solid.

3. Same idea applies to both theyaxis and any

other vertical axis. You simply must solve each equation forxbefore you plug them into the integration formula. 5 ex.Using the washer method, nd the volume gen- erated by rotating the region bounded by the given curves about the specied axis. y=x3; y=x; x0; abouty= 5.(9742 )6 (same as last one except abouty=2.)

Using the washer method, nd the volume generated

by rotating the region bounded by the given curves about the specied axis. y=x3; y=x; x0; abouty=2.(2521 )7 NYTI:

1. Determine the volume of the solid obtained by

rotating the portion of the region bounded by y=3pxandy=x4 that lies in the rst quadrant about theyaxis.(51221 )8 If we rotate about a horizontal axis then the cross- sectional area will be a function ofx. If we rotate about a vertical axis then it will be a function of y.

2. Determine the volume of the solid obtained by

rotating the region bounded byy= 2px1 and y=x1 about the linex=1.(965 )9

3. Using the Washer method, nd the volume gener-

ated by rotating the region bounded by the given curves about the specied axis. y= (x1)1=2; y= 0; x= 5; abouty= 3. ( 24)10quotesdbs_dbs12.pdfusesText_18