act5_sol - Solu‘llomfi’ Name Activity 5 1 Find the...

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Unformatted text preview: Solu‘llomfi’ Name: Activity 5 1. Find the total length of the graph of the astroid So HWY: 1+ 4_:( Than A: ;l 78/3 OlX X3 ‘3 V0 \ u; l M 96$ 2 _ Q 2 [X J312 9/ —dx - o : x8+4"x l: 4 V3 (2) o X 12/} ’ . . X973 Tolal lb‘na'l'ln = 4”?) “1‘43, 2. Two electrons repel each other with a force that varies inversely as the square of the distance between them. One electron is fixed at the point (2,4), Find the work done in moving the second electron from (—'2, 4) to (1,4). F()(') = 1 ———.___ (ll—x)2 -ez ‘ x hex ll 1 d1 2 — J \ elm-x). HEW—M 3 —_ F—-——-o———-—»—-m————g ’ (4—Xj2 _ (d‘x)2 er) ‘2 8I 2 e l |___._.,,-*____ , : [ i] V 1 Q l 2 .L ' 4. 44 3 e L 3. (9.) Suppose t = 10, find the area of the surface formed by revolv- ing the graph of f(x) = % about the x—axis on the interval [1. t] and find the volume of the solid formed by revolving the region bounded by f(3) = i and x =1 and :c =t. ‘2 a “I — I ‘ ~+l L1 "' —? 1+ll1 J x‘f it"- i _1_ 31:1 d'l ' 21! I 1414+: dz - x 1? .x3- (b) Repeat question (a) for t = 100 Io {=10 :3 H - 2'“ S x4311 d: l X r “T —- .1 = “I V ‘ (1 lo) I0. I00 {:100 fl 9 1= 2'“ j J¥~+I (I; I at": __ V - “(“7133 Ion (c) Use the information in (a) and (b) to write the volume and the area as functions of variable 1. (Use V(t) to denote the volume and A(t) to denote the area) (d) Find the limits (if exit) of V(t) and A(t) as t approaches infinity. lion “(1— ) 5 1T M. M, 1 {am ‘l “law I: Note: This shape is called Gabriel’s Horn or Torricelli’s Trumpet, which can be filled by 7r cubic units of paint, while you need infinite number of square units of paint to cover the surface! 4. A torus is formed by revolving the graph of (a: — 2)2 + y2 = 1 about ‘15 y-axis. Find the surface area of the torus. ,2 l_ x-2 . «Hafiz: 4+ (X'Q) (3:,Jl-(X‘2)? L1 ,l’-(x-Q)t J '*(X'2)2 ‘ 1 3.9, 4-(x—2) 33 3 ’2 Am A = 2<2TI>5 “W“ ' ‘ 0‘" I Jpn-'2)“; 3 , = 4“ [X]“ .-: 41l(3‘|) : 5. Determine the center of mass for the region bounded by y = x3 and ‘ " y=x/§w ' '3 a / <r 5 M=J(,I}_X)dxz[_2xz_&]=_ a 3 t‘ O l?- I 3 l L NX: jk’l—xl’V‘ )(j—x~x)d)l = l J(x—x)d)< o ‘L 2 0 7. 9‘ :: 1(2‘. " )5.) = 5 Z 2 *I 0 Es. \ | 3/ I" a 2- x M3:Sx(«l§«xm\x : g” X” 0 5’ g ‘ = (3“ - i) = l 5 5 ° 5 ...
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