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Jim Lamb ers Math 2B Fall Quarter 2004- Le ture 9 Notes
These notes orresp ond to Se tion 6.3 in the text.
Volume by Shells
In the previous le ture we learned how we ould ompute the volumes of a solid S that lay b etween the planes x = a and x = b by integrating its ross-se tional A(x) area over the interval [a; b℄. Unfortunately, this te hnique of omputing volume \by sli es", where ea h sli e is a ylinder of in nitely small height, is only pra ti al if the ross-se tional area is relatively easy to determine. If this is not the ase, then we an instead try to determine whether the solid an b e approxi- mated by several on entri ylindri al shel ls. For example, supp ose that b > a 0, and we have a solid that an b e obtained by rotating the region b ounded by y = f (x) (where f (x) 0), x = a, x = b, and y = 0. Then, the volume of resulting solid an b e approximated by rst dividing the interval [a; b℄ into n subintervals of equal width x = (b a)=n, with the ith subinterval having endp oints xi 1 and xi , where xi = ix for i = 0 ; : : : ; n. For ea h subinterval, we denote the midp oint xi by (xi 1 + xi )=2. Then, we ompute the volumes of n shells of thi kness x = (b a)=n, height f (xi ), inner radius xi 1 , and outer radius xi , for i = 1 ; : : : ; n. The volume of the ith shell is Vi = 2 xi f (xi )x, and therefore the volume V of the entire solid is approximated by
V
Xn
i=
Vi =
Xn
i=
2 xi f (xi )x: (1)
Letting n! 1 , this approximation onverges to the exa t volume, with the summation onverging to the de nite integral
V =
Z (^) b
a
2 xf (x) dx: (2)
More generally, if the solid an b e viewed as a olle tion of on entri ylindri al shells of radius r (x) and height f (x), for a x b, the volume of the solid is given by
V =
Z (^) b
a
2 r (x)f (x) dx: (3)
If the solid is formed by rotating a region around the y -axis, then r (x) = x. However, a di erent r (x) must b e used if the solid is obtained by rotating a region around a di erent line. For example, if the line around whi h the region is rotated is x = , where must b e outside the interval [a; b℄, then r (x) = x if a, while r (x) = x if b.
Example 1 In the previous le ture, we used the dis metho d to ompute the volume of the solid obtained by revolving the region b ounded by y = 2 p x, y = 0, x = 0 and x = 1 around the y -axis. Now, we will ompute the volume of the same solid using the shell metho d. The average radius of the shell at x is equal to x, and the height of this shell is equal to the height of the region at x, whi h is 2
p x. It follows that the volume of the solid is Z (^1)
0
2 x(2
p x) dx = 2
Z 1
2 x x^3 =^2 dx = 2 x^2 2 x^5 =^2 5
Example 2 Let f (x) = x^2. Consider the region b ounded by the urve y = f (x), the horizontal line y = 0, and the verti al lines x = 1 and x = 2. Compute the volume of the solid obtained by revolving this region around the y -axis.
Solution We use the metho d of ylindri al shel ls. The given region is shown in Figure 1, while the solid obtained by revolving the region around the y -axis is shown in Figure 2. This solid an b e approximated by a olle tion of on entri ylindri al shells. The approximation pro eeds as follows: rst, we divide the interval [1; 2℄ into n subintervals of equal width x = 1 =n. These intervals have endp oints [x 0 ; x 1 ℄, [x 1 ; x 2 ℄, : : : , [xn 1 ; xn ℄ where xi = 1 + ix, for i = 0 ; 1 ; 2 ; : : : ; n. Then, we approximate the region b elow y = f (x) by re tangles of width x and height f (xi ), where xi is any p oint in the interval [xi 1 ; xi ℄ for i = 1 ; : : : ; n. Then, by revolving ea h re tangle around the y -axis, we obtain n ylindri al shells that ap- proximate the solid, just as the re tangles approximate the region. This pro ess of revolving ea h re tangle around the y -axis to obtain a shell is illustrated in Figure 3. For ea h i = 1 ; : : : ; n, the ith shell has thi kness x, height f (xi ), inner radius xi 1 and outer radius xi. The volume Vi of this shell is given by
Vi = 2
xi 1 + xi 2
f (xi )x: (5)
By adding the volume of all of these shells, we obtain a Riemann sum that yields an approximation to the volume of the solid. As the numb er of subintervals, n, b e omes in nite, this approximation onverges to the exa t volume. The volume V of the solid is therefore given by
V = (^) nlim!
Xn
i=
Vi
= lim n!
Xn
i=
xi 1 + xi 2
f (xi )x
= (^) nlim!
Xn
i=
xi
x 2
f (xi )x
−
−1.
−
−0.
0
1
2
−
−1.
−
−0.
0
1
2
0
1
2
3
4
x
Solid of revolution
z
y
Figure 2: Solid obtained by revolving region shown in Figure 1 around the y -axis
Example 3 Let f (x) = x^2. Consider the region from the previous example, that is b ounded by the urve y = f (x), the horizontal line y = 0, and the verti al lines x = 1 and x = 2. Compute the volume of the solid obtained by revolving this region around the verti al line x = 2.
Solution In this ase, we an approximate the solid by ylindri al shells as b efore, but the enter and radii of the shells is di erent. Be ause the enter of the solid is the line x = 2, the inner radii of the ith shell, orresp onding to the subinterval [xi 1 ; xi ℄, is xi 1 + 2, sin e that is the distan e b etween the inner b oundary of the shell and the enter. Similarly, the outer radius of the ith shell is xi + 2. Pro eeding in the previous example, we an determine that the volume V of the solid is given by
V = (^) nlim!
Xn
i=
Vi
−1.5 (^) −1 (^) −0.5 (^0) 0.5 (^1) 1.5 2 −1.
−
−0.
0
1
0
1
2
3
4
z
x
Shell method
y
Figure 3: Cylindri al shell obtained by revolving re tangle around the y -axis. The re tangle is one of several that is used to approximate the region b ounded by y = x^2 , y = 0, x = 1, x = 2, whose outline is shown.
= (^) nlim!
Xn
i=
(xi 1 + 2) + (xi + 2) 2
f (xi )x
= lim n!
Xn
i=
xi + 2 x 2
f (xi )x
Z 2
2 (x + 2)f (x) dx: (8)
Evaluating the resulting de nite integral yields
V =
Z 2
2 (x + 2)x^2 dx
Z 2
x^3 + 2 x^2 dx
(^1) 1.
1.6 (^) 1. 2 −
−
0
2
4
−
−
−
−
0
1
2
3
4
z
Washer method
x
y
y=−
y=x^2
x=1^ x=
y=
Figure 4: The region b ounded by y = x^2 , y = 0, x = 1 and x = 2 is to b e revolved around the line y = 1. The washer orresp onding to x = 1 : 4 is shown.
Using the shell metho d, we integrate with resp e t to y b e ause we are revolving the region around a horizontal line. For ea h y in the interval [1; 4℄, we have a ylindri al shell entered at the line y = 1 with thi kness dy , average radius y + 1, and height 2 p y , sin e that is the horizontal distan e b etween the line x = 2 and the urve y = x^2 , or, equivalently, x = py. Furthermore, for ea h y in the interval [0; 1℄, we have a ylindri al shell entered at the line y = 1 with thi kness dy , average radius y + 1, and height 1. It follows that the volume V is given by
V =
Z 1
2 (y + 1)(1) dy +
Z 4
2 (y + 1)(2 p y ) dy
Z 1
y + 1 dy +
Z 4
2 y y 3 =^2 y 1 =^2 + 2 dy
y 2 2
y 5 =^2 5 = 2
y 3 =^2 3 = 2
2 45 =^2
2 43 =^2
2 15 =^2
2 13 =^2
A sample shell is illustrated in Figure 5. 2
