Volumes by Integration

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