6.3: Volumes by Cylindrical Shells

Transcription

1 6.3: Vlumes by Cylindrical Shells Nt all vlume prblems can be addressed using cylinders. Fr example: Find the vlume f the slid btained by rtating abut the y-axis the regin bunded by y = 2x x B and y = 0. The regin lks like this: Slicing this regin parallel t the x-axis and rtating abut the y-axis prduces washers whse radii vary. In the previus sectin we wuld have nted that the axis f rtatin determines the limits f integratin; in ther wrds, if yu rtate abut the x-axis yu integrate in terms f x, and if yu rtate abut the y-axis yu integrate in terms f y. In this case thugh, althugh we are rtating abut the y-axis, if we try t integrate with respect t y, we wuld have t integrate a functin in terms f x. We can prceed, nly if we can write the functin in terms f y. With a functin like y = x, that's easy, but y = 2x x B is harder t d this with. In fact, fr sme functins it's impssible. Instead, even thugh we are rtating arund the y-axis, we will "slice" the regin in the directin perpendicular t this, meaning that rather than washers btained by subtracting the inner circumference frm the uter

2 washers btained by subtracting the inner circumference frm the uter circumference, we will have washers whse heights vary. Each ne f ur washers will cme frm a different sample pint x R r xt R and will have a height f(x R r f(xt. R These "washers" are really t tall t be called washers, but they are hllw, s we can't really call them cylinders either. We call them "cylindrical shells". The vlume f ne cylindrical shell is the vlume f what wuld be the cylinder minus the vlume f the cylinder in the middle - yu can think f it as having remved a cre cylinder frm a larger ne.

3 The vlume f the uter cylinder is V = πr h where r is the radius f the uter cylinder. In ur diagram, this uter radius is labeled r, s we rewrite ur vlume frmula as V = π(r h. The vlume f the inner cylinder is similarly btained with V = π(r h. The vlume f the shell is the difference f these, r V = π(r h π(r h r V = πh[(r (r ]. With sme algebra magic, we can rewrite this frmula a different way: V = πh[(r (r ] = πh(r + r (r r = πh(r + r (r r = 2 ( r + r πh(r 2 r = 2π ( r + r h(r 2 r Nw ntice that r r is the difference between the tw radii, and s we can represent this by Δr and get: V = 2π ( r + r h Δr 2 We can als ntice that ( d e fd g is the average radius f the shell, and replace the expressin with r, which gives us: V = 2πrh Δr. In fact, each representative r is ne f ur x-values, s ultimately we will replace r with x. Ntice t, that the height h f each cylindrical shell is the value f the

4 Ntice t, that the height h f each cylindrical shell is the value f the functin fr r R, r f(r R. If this frmula is the vlume f ne cylindrical shell, and what we want is a sum f vlumes f several shells, then we replace r with x, and h with f(x, and integrate i h 2πxf(x j This frmula can be used fr any functin when the apprach being taken is that f cylindrical shells. The thing ne must be careful with is what "x" and "f(x" really mean... Let's use this frmula t finish the questin we started at the beginning f these ntes: Example: Find the vlume f the slid btained by rtating abut the y-axis the regin bunded by y = 2x x B and y = 0. Answer: h 2πx[2x x B ] = 2π h 2x B x p = 2π qh 2x B h x p = 2π q2 h x B h x p r r

5 = 2π q2 h x B h x p r = 2π s2 xp u 4 xv u x = 2π s xp u 2 xv u x = 2π yz 2p 0p { z 2v 0v { u 2 2 = 2π y(8 z 32 { u = 2π yz 40 { z 32 { u = 2π ~ = 2π ~ 8 = 16π Example: Find the vlume f the slid btained by rtating abut the y-axis the regin between y = x and y = x.

6 Answer: Using cylinders (sectin 6.2 we wuld need t use the frmula i V = h A(x j where A(x = πr πr = πƒ( y (y = π(y y which gives us V = h π(y y V = π h y y dy V = π z y yb {u 2 3 V = π z { V = π 6 dy Using shells, ur frmula is

7 i V = h 2πxf(x j where f(x = x x s V = h 2πx(x x V = 2π h x(x x V = 2π h x x B V = 2π y xb xp u 3 4 V = 2π yz 1B 1p { z 0B 0p {u V = 2π z 4 { 3 12 V = π 6 There are times when yu can chse t use either methd. In fact, it can be argued that yu culd always use either methd. The chice cmes dwn t which is easier.

8 Dn't frget that f(x is the area f the base f a cylinder fund by taking a crss sectin parallel t the axis f rtatin. Hmewrk: 1, 3, 13, 17