N MATH132O: hitegration Techniques and Area Between Curves Iustnictor: Laura Strube 26 August 2015 Name: Student ID: (A 1. Integration by Parts Integrate: eç f’ 2 r dr using the integration by parts method. 2 4 V 1nd V V _— A LÀ (-u-o r Ow=2rch’ r 2r - *r2 - SP2 [4-tt b P&s - S2 u &ubsf#H dU2rcIr = 2 1 )21 I ii JfD (kr2) L + L-J >c;} + x N ‘I 2 + N -f U >N LS -I N ‘I 0 ‘I t’J + I’ Ii, N I’ N I\J N 4 ii I ‘I - xl”) >( Js) -I \Lfr ÷ ‘I N I x t’J + -I •Ii’ + >c N ‘ -I- N >c N N ‘I I”) 4- >c r. -I >c 4 ‘I N >s + Li 4. Xty: N vJ g) t1 ii p.’ U, I — CD CD 0 CD CD ,-. o- — cn 0 JCiD cr o zQ — - ZCD .: 3. Area under a Curve . Begin by sketching the 2 and g() = 2x r Find the area between the curves f(x) = 2 to determine the limits of integration. curves and solving x’ = 2x x — — ()(): 2 ZX -)( + ((Z_ - Zx _(_2x+ —‘ bonur cir’ rL (YIZ (A) b-’- ck 0 or W - Cx) 2.X- Zx —X 2 x I : Eca j) (x-*--2’) =0 Ix ( -4-1-- )( ‘1.- iX2 - x’ -4- — 0 Dcnn+ -+X+2. 2 X ‘t-eQ i rz,cDi3 4. Integrating with Respect to y 2 = 4 and the line 4x 3y 4 by integrating with Find the area between the parabola y 2 3y + 4 to determine the respect to y. Begin by sketching the cnrves and by solving y limits of integration. 2— phi IA U ::.-X 2 A x - _.L Limsc Tnfcfl 2 c: 3i-+ - - ‘I— 1_. 1 =-I4 ‘o1n j- / Not 4 x - 3 -4 jkI 3j .jt 41 l 4 i-I /L--_I’) (4) — 2 L_j’) 2 0 5. Simpson’s Rule and Velocity Curves The following figure shoes velocity curves for two cars, A and B, that start side by side and move along the same road. \Vhat does the area between the curves represent? Use Simpson’s Rule to estimate it. (A table of function values is provided to simplify the calculation) ( jc,ryipe Or’ 434 3 d ur 1. 09 £9 6Z 6 9c cz ic LI frL 0 0 -_____-___ 68 c6 I t’8 0i iyl 9 8 L9 9L tc IL 0 0 3flN (sp uo s)