Volumes by Cylindrical Shells

1 Some volume problems are very difficult to handle by the methods of Section Forinstance, let s consider the problem of finding the volume of the solid obtained by rotatingabout the -axis the region bounded by and . (See Figure 1.) If we sliceperpendicular to the y-axis, we get a washer. But to compute the inner radius and the outerradius of the washer, we would have to solve the cubic equation for xinterms of y; that s not , there is a method, called the method of Cylindrical Shells , that is easier touse in such a case. Figure 2 shows a Cylindrical shell with inner radius , outer radius ,and height . Its volume is calculated by subtracting the volume of the inner cylinderfrom the volume of the outer cylinder:If we let (the thickness of the shell ) and (the average radiusof the shell ), then this formula for the volume of a Cylindrical shell becomesand it can be remembered asNow let be the solid obtained by rotating about the -axis the region bounded by[where ],and , where.

2 (See Figure 3.) We divide the interval into nsubintervals of equal width and let bethe midpoint of the ith subinterval. If the rectangle with base and height isrotated about the y-axis, then the result is a Cylindrical shell with average radius , height, and thickness (see Figure 4), so by Formula 1 its volume isTherefore, an approximation to the volume of is given by the sum of the Volumes ofthese Shells :V ni 1 Vi ni 1 2 xif xi xSVVi 2 xi f xi x xf xi xif xi xi 1, xi xi x xi 1, xi a, b FIGURE 3abxy0y= xyab0y= b a 0x by 0, x a, f x 0y f x ySV [circumference][height][thickness]V 2 rh r1r 12 r2 r1 r r2 r1 2 r2 r12 h r2 r1 r2 r1 r2 r1 h r22h r21h r22 r21 h V V2 V1V2V1 Vhr2r1y 2x2 x3y 0y 2x2 x3yFIGURE 1 FIGURE 2rr r rhyx021y=2 - xL=?xR=?1 Volumes by Cylindrical ShellsThis approximation appears to become better as . But, from the definition of an inte-gral, we know thatThus, the following appears plausible:The volume of the solid in Figure 3, obtained by rotating about the y-axis theregion under the curve from ato b,isThe argument using Cylindrical Shells makes Formula 2 seem reasonable, but later wewill be able to prove it.

3 (See Exercise 47.)The best way to remember Formula 2 is to think of a typical shell , cut and flattened asin Figure 5, with radius x, circumference , height , and thickness or :This type of reasoning will be helpful in other situations, such as when we rotate aboutlines other than the 1 Find the volume of the solid obtained by rotating about the -axis the regionbounded by and .SOLUTIONFrom the sketch in Figure 6 we see that a typical shell has radius x, circumfer-ence , and height . So, by the shell method, the volume isIt can be verified that the shell method gives the same answer as 7yx 2 [12x4 15x5]02 2 (8 325) 165 V y20 2 x 2x2 x3 dx 2 y20 2x3 x4 dxf x 2x2 x32 xy 0y 2x2 x3yFIGURE 52 x x yxxx heightcircumferencedx f x 2 x ybadx xf x 2 xwhere 0 a b V yba 2 xf x dxy f x 2lim nl ni 1 2 xif xi x yba 2 xf x dxnl 2 Volumes BY Cylindrical SHELLSFIGURE 4xyab0y= xi ab0xyxi-1xiy= FIGURE 6yx2 - xx2 Figure 7 shows a computer-generated picture of the solid whose volume we computedin Example Comparing the solution of Example 1 with the remarks at the beginning of thissection, we see that the method of Cylindrical Shells is much easier than the washer methodfor this problem.

4 We did not have to find the coordinates of the local maximum and we didnot have to solve the equation of the curve for in terms of . However, in other examplesthe methods of the preceding section may be 2 Find the volume of the solid obtained by rotating about the -axis the regionbetween and .SOLUTIONThe region and a typical shell are shown in Figure 8. We see that the shell hasradius x, circumference , and height . So the volume isAs the following example shows, the shell method works just as well if we rotate aboutthe x-axis. We simply have to draw a diagram to identify the radius and height of a 3 Use Cylindrical Shells to find the volume of the solid obtained by rotatingabout the -axis the region under the curve from 0 to problem was solved using disks in Example 2 in Section To use shellswe relabel the curve (in the figure in that example) as in Figure 9.

5 Forrotation about the x-axis we see that a typical shell has radius y, circumference , andheight . So the volume isIn this problem the disk method was 4 Find the volume of the solid obtained by rotating the region bounded byand about the line .SOLUTIONF igure 10 shows the region and a Cylindrical shell formed by rotation about theline . It has radius , circumference , and height .The volume of the given solid is 2 x44 x3 x2 01 2 V y10 2 2 x x x2 dx 2 y10 x3 3x2 2x dxFIGURE 100yxy=x- 0yxx12342-xx=2x x22 2 x 2 xx 2x 2y 0y x x2 2 y22 y44 01 2 V y10 2 y 1 y2 dy 2 y10 y y3 dy1 y22 yx y2y sxy sxx 2 x33 x44 01 6 V y10 2 x x x2 dx 2 y10 x2 x3 dxx x22 xy x2y xyyxVOLUMES BY Cylindrical Shells 3 FIGURE 80xyy=xy= xshellheight=x- FIGURE 91yyshellradius=yshell height=1- 0xx=11x= 4 Volumes BY Cylindrical be the solid obtained by rotating the region shown in the figure about the -axis.

6 Explain why it is awkward to useslicing to find the volume of . Sketch a typical approxi-mating shell . What are its circumference and height? Use shellsto find . be the solid obtained by rotating the region shown in thefigure about the -axis. Sketch a typical Cylindrical shell andfind its circumference and height. Use Shells to find the volumeof . Do you think this method is preferable to slicing? 7 Use the method of Cylindrical Shells to find the volume gen-erated by rotating the region bounded by the given curves about the-axis. Sketch the region and a typical ,,,4.,, ,7., be the volume of the solid obtained by rotating about the-axis the region bounded by and . Find bothby slicing and by Cylindrical Shells . In both cases draw a dia-gram to explain your 14 Use the method of Cylindrical Shells to find the volume ofthe solid obtained by rotating the region bounded by the givencurves about the -axis.

7 Sketch the region and a typical .,,12.,13.,14. x y 3,x 4 y 1 22x y 6y 4x2x 0x 4y2 y3x 0y 8y x3x sy,x 0,y 1x 1 y2,x 0,y 1,y 2xVy x2y sxyVy x2 4x 7y 4 x 2 2x y 3y 3 2x x2y e x2,y 0,x 0,x 1x 1y 0y x2x 2x 1y 0y 1 xy0xy y=sin{ }SyS0xy1y=x(x-1)@VSVyS15 20 Use the method of Cylindrical Shells to find the volumegenerated by rotating the region bounded by the given curves aboutthe specified axis. Sketch the region and a typical ,;about 16.,;about the -axis17.,;about 18.,;about 19.,;about 20. 21 26 Set up, but do not evaluate, an integral for the volume of the solid obtained by rotating the region bounded by the givencurves about the specified ,;about 23.,;about the Midpoint Rule with to estimate the volumeobtained by rotating about the -axis the region under the curve,. the region shown in the figure is rotated about the -axis toform a solid, use the Midpoint Rule with to estimate thevolume of the 32 Each integral represents the volume of a solid.

8 Describethe y 40 2 x cos x sin x dxy10 2 3 y 1 y2 dy2 y20 y1 y2 dyy30 2 x5 dx0xy112345234567 89101112n 5y0 x 4y tan xyn 4y 5x2 y2 7,x 4;y 4x ssin y,0 y ,x 0;x 2y 1 1 x2 ,y 0,x 0,x 2;x 1y sin x 2 y x4x 7y 4x x2y xy ln x,y 0,x 2; about the y-axisy x2,x y2;about y 1y 3y 0,x 5y sx 1x 2y 8x 2x2y 4x x2x 4y 0,x 1,x 2y x2yy 0,x 2,x 1y x2x 1y 0,x 1,x 2y x2 Click here for here for BY Cylindrical Shells 5;33 34 Use a graph to estimate the -coordinates of the points ofintersection of the given curves. Then use this information to esti-mate the volume of the solid obtained by rotating about the -axisthe region enclosed by these ,34., 35 36 Use a computer algebra system to find the exact volumeof the solid obtained by rotating the region bounded by the givencurves about the specified ,,;about 36.,,;about 37 42 The region bounded by the given curves is rotated aboutthe specified axis.

9 Find the volume of the resulting solid by ,;about the -axis38.,;about the -axis39.,;about 40.,;about 41.;about the -axis42.;about the -axis 43 45 Use Cylindrical Shells to find the volume of the sphere of radius solid torus (a donut-shaped solid with radii and )shown in the figurerRrRrxx2 y 1 2 1yx2 y 1 2 1x 2x 0x 1 y4x 1y x 4 x y 5yy 0y x2 3x 2xy 0y x2 x 2x 10 x y 0y x3 sin xx 20 x y sin4xy sin2xCASy 3x x3y x4y x x2 x4y right circular cone with height and base radius you make napkin rings by drilling holes with differentdiameters through two wooden balls (which also have differentdiameters). You discover that both napkin rings have the sameheight , as shown in the figure.(a) Guess which ring has more wood in it.(b) Check your guess: Use Cylindrical Shells to compute thevolume of a napkin ring created by drilling a hole withradius through the center of a sphere of radius andexpress the answer in terms of.

10 Arrived at Formula 2,, by usingcylindrical Shells , but now we can use integration by parts toprove it using the slicing method of Section , at least for thecase where is one-to-one and therefore has an inverse func-tion . Use the figure to show thatMake the substitution and then use integration byparts on the resulting integral to prove that .y0xabcdx=ax=by= x=g( y)V xba 2 xf x dxy f x V b2d a2c ydc t y 2 dytfV xba 2 xf x dxhhRrhrh6 Volumes BY Cylindrical , height; 1 1 e xy01xxyx2 x(x-1)@0yx1xyx1 15 x x 1 2 2 .13 . obtained by rotating the region ,about the obtained by rotating the region bounded by (i) ,, and , or (ii) ,, and about the line , ; r2h43 r34 38 3 ln 4 81 10132 3y 3y 0x 1x y2y 0x 0x 1 y2y0 x 30 y x4x 0 2 4 y ssin y dyx10 2 x 1 sin x 2 x4 dxx21 2 x ln x dx24 67 617 6250 3768 7xy011+ 12xy01y21 2 Click here for we were to use the washer method, we would first haveto locate the local maximum point(a, b)ofy=x(x 1)2using the methods of Chapter 4.