MH AY67 Sem Questio. NOT TESTED THIS TIME ( marks Let R be the regio bouded by the curve y 4x x 3 ad the x axis i the first quadrat (see figure below. Usig the cylidrical shell method, fid the volume of the solid geerated by revolvig the regio R about the vertical lie x 3. y R y 4x x 3 x 3 Solutio. V π π(3 x(4x x 3 dx x x 3 + x 4 dx [ π 4x 3 5 x4 + ] 5 x5 π (3 5 (6 + 5 (3 48 5 π.
Questio. ( marks Evaluate the followig itegrals. Express your fial aswers i terms of x. 5x x 3 x + 4x 4 dx (b 3 x x dx Solutio. The 5x x 3 x + 4x 4 5x x (x + 4(x 5x A x + Bx + C x + 4. (x (x + 4 5x A(x + 4 + (Bx + C(x (A + Bx + (C Bx + 4A C. A, B C 4. Thus, 5x x 3 x + 4x 4 dx x + x dx + 4 x + 4 dx x l x + 4 x + 4 dx + 4 x + 4 dx l x + l x + 4 + 4 4 ta θ + 4 sec θ dθ l ( x x + 4 ( x + ta + C. (b By completig squares, 3 x x 4 (x +.
Set x + si θ, where π/ θ π/. The dx cos θ dθ. Thus, 3 x x 4 4 si θ cos θ dθ 4 cos θ dθ 4 (cos θ + dθ ( si θ + θ + C si θ cos θ + θ + C (x + ( 3 x x x + + si + C. 3
Questio 3. Determie whether the followig series coverges or diverges. Justify your aswer. (b (c Solutio. 3 6 5 + 7 4 6 + + 4 + 5 5 + 6 l + 5 3 6 5 + 7 4 6 + + 3 5 4 + 7 6 4 + 5 + 6 3 4. By the -th Test for Divergece, the series diverges. (b Let a 4 +5 5 +6, b 5 6. The a 4 + 5 b 5 + 6 6 5 ( 4 5 + ( 5. 6 + The geometric series ( 5 6 coverges. By the Limit Compariso test, the series a also coverges. Let a l +5, ad let b 3/. Sice the p-series series coverges. a b l + 5 3/ l + 5. a lim lim b 3/ /x x x x / lim x. coverges, By the Limit Compariso Test, the 4
Questio 4. Use power series to evaluate e x e x lim x si 3x. (b Fid the iterval of covergece of the followig power series, ad idetify the values of x for which the series coverges absolutely or coditioally. (4x + 3 +. 3 + Solutio. e x e x lim x si 3x ( + x + x + ( x + x!! lim x (3x (3x3 + 3! ( x + x3 + x5 + 3! 5! lim x (3x (3x3 + 3! ( + x +! lim 3. x 3 (3x! + (b Let a (4x+3+ 3+. The a + a 4x + 3 + 3 + 4 3 + 3 + 4x + 3 4x + 3. 4x + 3 + 3 + 4 By ratio test, the series is coverget if 4x + 3 < < x <. 5
If x, the a 3 + which diverges sice it is a Harmoic series. If x, the ( a 3 + which coverges by the Alteratig Series Test. 6
Questio 5. Suppose lim a L, where L > is a real umber. Show that the power series a x coverges for x < L. Solutio. Use Root Test. lim a x lim a x L x. Thus, the series coverges if L x <, i.e. x < L. Alterative solutio. Suppose x <. Let r be a fixed real umber such that L < r <. The L x rx <. Sice a L, there exists N such that a < r for all N. Thus, for all N, we have a < r. Hece a x < rx. The series rx is a coverget geometric series. So, by Compariso Test, the series a x coverges. Hece, the series a x coverges (absolutely. 7
Questio 6. ( marks Fid the Taylor series cetred at c for the fuctio f(x, ad determie its radius of covergece. 3x (b Let F (x Solutio. π/ x si t dt. Fid the Maclauri series for F (x. 3x 3(x + 3(x + 5 5 3(x + ( 5 3 (x + 5 ( 3 (x + 5 5 By ratio test, the series coverges o the iterval ( 3 + 5 (x + + ( 3 5 (x + < The radius of covergece is 5 3. x + < 5 3 (b Usig the biomial series expasio, we have, for y <, ( + ( y / 8 ( ( y.
Substitutig y x si t, we have for x < ( x si t <, ( ( x si t / ( x si t. ( Let I π/ (si t dt. By itegratio-by-parts, I [ cos t si t ] π/ ( π/ π/ + ( si t si t dt cos t( si t dt ( I ( I I I for all. ( Thus, I 3 I 4. (3 3 I 3 π Itegratig (, we have π/ F (x x si t dt π/ ( ( x si t dt for all. (4 π/ ( π/ ( x si t dt π π/ + ( ( x I π + π π/ ( ( 3 x (5 9
Set A( 3. The F (x π + π π/ ( ( A( x (6 Recall that ( ( ( ( ( (! ( ( 3( 5 ( ( 3! 3 5 ( ( (! Substitutig this ito (5, we have 3 5 ( 3 (! ( A(. F (x π + π π/ ( A(( A( x π π π/ A( x.