Question

Q3 ) Use numerical integration with n-3 to determine the following integration then solve it analytically using a proper method of integration and compare results. InLX dx 04) Used double integral find the area between the following curves y = x2y = 2 - x (5) Use shell method find the volume of the solid generated by revolving the curves y=x", y = 2 - x?, about y axis

Answer 1

VPAGE NO.: DATE: 1 #SOLUTION + 3 1 using Numerical integration ( Trapezoidal Rule) no 3 he upper limit - Lower Limit 3-1 = 2 3 n 3 & fi= f(xi) Inxi ber= atih xi then trapezoidal Rule. b fix) dx = h {f%) + 2 (1+2 9 here n=3, then 3 Inx dx F h 2 2 {(x+42 +2 (4+4) 1 finding to Xe = 1 Y In L=0 + 1 3 In(5/3) 5 Xg = 1 +h = 1+ = { → Y, = ln (513) 513 X₂ = 1+2h = 1 +2 (3) = = 2 7 7 Y = ln (713) 3 ln (7 7. 3 713

x3 = 1+3h = 1+3(3) = 3 PAGE NO.: DATE: 1 Y = ln (3) 3 therefore, 3 fox dx = / h 4 (Y+Y) 42 (9,4%) Ho{font 1,3). 42 (223)+2(43); ulto) In 3 + (Heft fi foarte 6 5 Inc 7 3 3 7 1.7055 3 Inx.di 0.5685 Ans. x Analytical method ; 0.6035 Ans.

POR NO.1 DATE y = x² y = 2-12 about y-axis sketching Regim Region that is Rotating 1 q=2-24 finding inter ser min points y = x2 2-72 = 72 2 - 2 X2 202 = 1 X = +1 aboret volume of solid generated by revolving is given by y-axis lott Z fx-914) y dx b = 217 X X=9 where fex *upper function g(x) lower funchin 2-X2 ( here) then v = att x ( 2-x*-x2) dx

PAGE NO.: DATE: 1 V = 2t * (2-2x2) dx att ( 2x 273 ) dx 0 = 2t 2 x dx x3 dx o = 21 L 72 # 1X4 4 picha = 21 (1-0) -1 (1-0) } = 21163 = 21 (1-3) 16402 biloz - 2 17 sxe V = tt Volume E IT