Instructor Hentzel Office Phone: 515-294-8141 E-mail: hentzel@iastate.edu Math Department Fax: 515-294-5454 http://www.math.iastate.edu/hentzel/class.166.ICN Textbook: Calculus Early Vectors Preliminary Edition By Stewart. Fri, Feb 14 8.1 Integration by Parts. p463: 1,4,(6),(18),14,17,22,23,27,30,32,33,37,38,41,(44),43,46,47,50,51 u --- v |\ /| | \ / | Main Idea: INT u dv = | X | = u v - INT v du | / \ | |/ \| du ---dv Key Words: INT u dv = uv - INT v du. Integration by Parts. Goal: Learn a way to simplify problems so that one can integrate them. ------------------------------------------------------------- Previous Assignment p444: (1),5,(10),12,16,19,(22),26,27,30,33, 38,(46) Page 444 Problem 1 Use the method of cylindrical shells to find the volume generated by rotating the region bounded by the given curves about the y-axis. y = x^2, y = 0, x=1, x=2. a = Plot[x^2,{x,1,2}]; b = Plot[0,{x,1,2}]; c = ParametricPlot[{1,y},{y,0,1},PlotStyle->{Thickness[0.01]}]; d = ParametricPlot[{2,y},{y,0,4}]; e = Show[a,b,c,d,PlotLabel->"Page 444 Problem 1"]; Display["1.ps",e]; a = ParametricPlot3D[{r Cos[t],r Sin[t],r^2},{r,1,2},{t,0,2 Pi},PlotPoints->50]; b = ParametricPlot3D[{2 Cos[t],2 Sin[t],z},{t,0,3/2 Pi},{z,0,4}]; c = ParametricPlot3D[{1 Cos[t],1 Sin[t],z},{t,0,2 Pi},{z,0,1}]; x=2 INT 2 Pi x x^2 dx x=1 x=2 2 Pi INT x^3 dx = 2 Pi | x^4/4 | = 15 Pi/2 <== Answer x=1 Check using coins. y=4 Volume = Pi (2^2) 1 - Pi 1^1 1 + INT Pi(2^2 - (Sqrt[y])^2 ) dy y=1 y=4 |y=4 Volume = 3 Pi + Pi INT 4 - y dy = 3 Pi + Pi |( 4y -y^2/2) | y=1 | |y=1 Volume = 3 Pi + Pi(16-8-(4-1/2)) = 15 Pi/2 ----------------------------------------------------------------------- Page 444 Problem 11 Use the method of cylindrical shells to find the volumn of the solid obtained by rotating the region bounded by the given curves about the x-axis. Shell: y = x^2, y=9 a = Plot[x^2,{x,-3,3}]; b = Plot[9,{x,-3,3}]; c = Show[a,b,PlotLabel->"Page 444 Problem 10"]; Display["10.ps",%] a = ParametricPlot3D[{r Cos[t],r Sin[t],r^2},{t,0,3/2 Pi},{r,0,3},PlotPoints->50]; b = ParametricPlot3D[{r Cos[t],r Sin[t],9},{t,0,2 Pi},{r,0,3},PlotPoints->50]; a=ParametricPlot3D[{x,x^2 Cos[t],x^2 Sin[t]},{x,-3,3},{t,Pi,2Pi},PlotPoints->50]; b = ParametricPlot3D[{x, 9 Cos[t],9 Sin[t]},{x,-3,3},{t,Pi,2 Pi},PlotPoints->50]; y=9 | | y=9 INT 2 Pi y (2 Sqrt[y]) dy = 4 Pi INT y^(3/2) dy = 4 Pi | y^(5/2)/(5/2) | y=0 | | y=0 8/5 Pi (243) = 1944 Pi/5. <=== Answer ------------------------- Coin: x=3 | x=3 INT Pi (9^2 - (x^2)^2) dx = Pi INT 81 - x^4 dx = Pi (81x - x^5/5) | x=-3 | x=-3 = Pi(243 -243/5 - (-243 +243/5) ) = 243 Pi (8/5) = 1944 Pi/5. <== Answer ======================================================================= Page 444 Problem 22 Set up but do not evaluate, an integral for the volume of the solid obtained by rotating the region bounded by the given curves about the specified axis. y = 1/(1+x^2) y = 0, x = 0, x = 3; about the y-axis. a = Plot[1/(1+x^2),{x,0,3}]; b = Plot[0,{x,0,3}]; c = ParametricPlot[{0,y},{y,0,1}]; d = ParametricPlot[{3,y},{y,0,1/10}]; e = Show[a,b,c,d,PlotLabel->"Page 444 Problem 22"]; Display["22.ps",e]; y=1 Coin INT Pi (1/y - 1) dy + Pi 3^2 1/10 y=1/10 | | y=1 | Pi ln(y) - Pi y | + 9/10 Pi | | y=1/10 0 - Pi -(Pi ln(1/10) - Pi/10) + 9/10 Pi -Pi + Pi ln(10) + Pi/10 + 9/10 Pi Pi ln(10) <===== Answer --------------------------------------- Shell: | | x=3 x=3 | ln(1+x^2) | INT 2 Pi x (1/(1+x^2)) dx = 2 Pi|----------- | x=0 | 2 | x=0 = Pi (ln(10) - ln(1)) = Pi ln(10). <==== Answer ================================================================== Page 444 Problem 46 If the region shown in the figure is rotated about the y axis to form a solid, use the Midpoint rule with n=5 to estimate the volume of the Solid. (* Working the problem as the book suggested *) x 3 5 7 9 11 f(x) 2 4 4 2 1 x=12 INT 2 Pi x y dx x=2 V = 2 Pi( 3 2 + 5 4 + 7 4 + 9 2 + 11 1 )*2 = 332 Pi = 1043.01 (* Working the problem using Riemann sum with more data points *) x 2, 3, 4, 5, 6, 7, 8, 9, 10, 11 12, f(x) 0 2 2.6 4 4.4 4 2.5 2 1,5 1 0 V = 2 Pi (3 2 + 4 2.6 + 5 4 + 6 4.4 + 7 4 + 8 2.5 + 9 2 + 10 1.5 + 11 1) V =972.637 (* Working the problem using the coin method. y=4 INT Pi (xr)^2 - Pi (xl)^2 dy y=0 y 1 2 3 4 4 3 2 1 x 11 9 7.6 7 5 4.4 3 2.2 V = Pi( 11^2 + 9^2 + 7.6^2 + 7^2 -5^2 -4.4^2 - 3^2 - 2.2^2 ) = 787.157 (* Working the problem by approximating the curve with a polynomial *) 2 3 4 p(x)= -121.818 + 177.626 x - 105.054 x + 33.2233 x - 6.13176 x + 5 6 7 8 0.680107 x - 0.0446142 x + 0.00159391 x - 0.0000238878 x Integrate[2 Pi x p[x], {x,2,12}] = 985.298 ================================================================== Theory: d(u v) = u dv + v du. Therefore uv = INT u dv + INT v du Therefore INT u dv = uv - INT v du. Page 459 Example 1 INT x Sin[x] dx = -x Cos[x] - INT -Cos[x] dx = -x Cos[x] + Sin[x] + C x -Cos[x] dx Sin[x] dx Page 460 Example 2 INT ln(x) dx = x ln(x) - INT dx = xln(x) -x +C ln x x 1/x dx dx Page 460 Example 3 INT x^2 e^x dx = x^2 e^x - INT 2 x e^x dx = x^2 e^x -2 x e^x + INT 2 e^x dx = x^2 e^x -2 x e^x + 2 e^x + C x^2 e^x 2 x e^x 2 x dx e^x dx 2 dx e^x dx Page 461 Example 4 INT e^x Sin x dx = -e^x Cos[x] + INT e^x Cos[x] dx INT e^x Sin x dx = -e^x Cos[x] + e^x Sin[x] - INT e^x Sin[x] dx e^x -Cos[x] e^x Sin[x] e^x dx Sin[x] dx e^x dx Cos[x] dx 2 INT e^x Sin[x] dx = e^x( -Cos[x] + Sin[x] ) INT e^x Sin[x] dx = (1/2) e^x( -Cos[x] + Sin[x] ) Check. (1/2) e^x( -Cos[x] + Sin[x] ) (1/2) e^x( Sin[x] + Cos[x] ) ----------------------------- e^x Sin[x] it checks. Page 462 Example 5 x=1 x INT ArcTan[x] dx = x ArcTan[x] - INT ---------- dx x=0 1+x^2 | x=1 = x ArcTan[x] -1/2 ln(1+x^2) | | x=0 = Pi/4 -1/2 ln(2) - (0-0) = Pi/4 -ln(2)/2. ArcTan[x] x dx ---------- dx 1+x^2 Page 462 Example 6 Create the reduction formula to integrate INT Sin[x]^n dx. INT Sin[x]^n dx = -Sin[x]^(n-1) Cos[x] +(n-1) INT Sin[x]^(n-2) Cos[x]^2 dx INT Sin[x]^n dx = -Sin[x]^(n-1) Cos[x] +(n-1) INT Sin[x]^(n-2) (1-Sin[x]^2) dx INT Sin[x]^n dx = -Sin[x]^(n-1) Cos[x] +(n-1) INT Sin[x]^(n-2) dx -(n-1) INT Sin[x]^2) dx Collection terms gives n INT Sin[x]^n dx = -Sin[x]^(n-1) Cos[x] + (n-1) INT Sin[x]^(n-2) dx n-1 INT Sin[x]^n dx = (-1/n) Sin[x]^(n-1) Cos[x] + ---- INT Sin[x]^(n-2) dx n Sin[x]^(n-1) -Cos[x] (n-1)Sin[x]^(n-2) Cos[x] dx Sin[x] dx