Josh Engwer (TTU) Area Between Curves 22 January / 66
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1 Area Between Curves Calculus II Josh Engwer TTU 22 January 2014 Josh Engwer (TTU) Area Between Curves 22 January / 66
2 Continuity & Differentiability of a Function (Notation) Definition Given function f (x) and set S R. Then: f C(S) f is continuous on set S f C 1 (S) f, f C(S) = f is differentiable on set S f C 2 (S) f, f, f C(S) = f is twice-differentiable on set S REMARK: In general, f being differentiable on set S may not imply that f C 1 (S). { x One such example is f (x) = 2 sin ( ) 1 x, if x 0 0, if x = 0 Such pathological functions will not be considered in this course. NOTATION: R Interval (, ) The set of real numbers The real line Josh Engwer (TTU) Area Between Curves 22 January / 66
3 Definite Integrals (Definition & Interpretation) Definition (Riemann Sum Definition of an Integral) Let f C[a, b] where [a, b] is a closed interval s.t. < a < b <. Then: b a f (x) dx := lim N k=1 N f (xk ) x Proposition (The Integral as an Area) Let f C[a, b] s.t. f (x) 0 x [a, b]. Then b a f (x) dx represents the area of the region bounded by the curve y = f (x), the x-axis, and the vertical lines x = a & x = b. Josh Engwer (TTU) Area Between Curves 22 January / 66
4 Riemann Sums (Non-Uniform, Arbitrary Tags) # Rectangles : N = 3 Width of k th Rectangle = x k Partition : P = {x 0, x 1, x 2, x 3 } Tags : T = {x1, x 2, x 3 N } Riemann Sum : f (xk ) x k = f (x1) x 1 + f (x2) x 2 + f (x3) x 3 k=1 Josh Engwer (TTU) Area Between Curves 22 January / 66
![Riemann Sums (Uniform, Left-Endpoint Tags) On the k th subinterval [x k 1, x k ],](/docs-images/95/122578772/images/5-0.jpg "let k th tag x k := x k 1.")
5 Riemann Sums (Uniform, Left-Endpoint Tags) On the k th subinterval [x k 1, x k ], let k th tag x k := x k 1. Josh Engwer (TTU) Area Between Curves 22 January / 66
6 Riemann Sums (Uniform, Right-Endpoint Tags) On the k th subinterval [x k 1, x k ], let k th tag x k := x k. Josh Engwer (TTU) Area Between Curves 22 January / 66
7 Riemann Sums (Uniform, Midpoint Tags) On the k th subinterval [x k 1, x k ], let k th tag x k := 1 2 [x k 1 + x k ]. Josh Engwer (TTU) Area Between Curves 22 January / 66
8 Riemann Sum Definition of an Integral (Demo) (DEMO) RIEMANN SUM DEFINITION OF AN INTEGRAL (Click below): Josh Engwer (TTU) Area Between Curves 22 January / 66
9 Boundary Curves (BC s) & Boundary Points (BP s) A region in R 2 has boundary points (BP s) & boundary curves (BC s). A boundary point (BP) is the intersection of two boundary curves (BC s). NOTATION: R 2 xy-plane. Josh Engwer (TTU) Area Between Curves 22 January / 66
10 Boundary Curves (BC s) & Boundary Points (BP s) Josh Engwer (TTU) Area Between Curves 22 January / 66
11 Vertically-Simple (V-Simple) Regions Definition A region D R 2 is vertically-simple (V-Simple) if the region has only one top BC & only one bottom BC. Josh Engwer (TTU) Area Between Curves 22 January / 66
![Vertically Simple (V-Simple) Regions (Definition) i.e., V-Simple regions can be swept vertically (with vertical lines [in blue])](/docs-images/95/122578772/images/12-0.jpg "where each vertical line intersects the same top BC & same bottom BC.")
12 Vertically Simple (V-Simple) Regions (Definition) i.e., V-Simple regions can be swept vertically (with vertical lines [in blue]) where each vertical line intersects the same top BC & same bottom BC. Josh Engwer (TTU) Area Between Curves 22 January / 66
13 Vertically-Simple (V-Simple) Regions Definition A region D R 2 is vertically-simple (V-Simple) if the region has only one top BC & only one bottom BC. Josh Engwer (TTU) Area Between Curves 22 January / 66
14 Vertically-Simple (V-Simple) Regions Definition A region D R 2 is vertically-simple (V-Simple) if the region has only one top BC & only one bottom BC. Josh Engwer (TTU) Area Between Curves 22 January / 66
15 Horizontally-Simple (H-Simple) Regions Definition A region D R 2 is horizontally-simple (H-Simple) if the region has only one left BC & only one right BC. Josh Engwer (TTU) Area Between Curves 22 January / 66
16 Horizontally Simple (H-Simple) Regions (Definition) i.e., H-Simple regions can be swept horizontally (w/ horizontal lines [in blue]) where each horizontal line intersects the same left BC & same right BC. Josh Engwer (TTU) Area Between Curves 22 January / 66
17 Horizontally-Simple (H-Simple) Regions Definition A region D R 2 is horizontally-simple (H-Simple) if the region has only one left BC & only one right BC. Josh Engwer (TTU) Area Between Curves 22 January / 66
18 Horizontally-Simple (H-Simple) Regions Definition A region D R 2 is horizontally-simple (H-Simple) if the region has only one left BC & only one right BC. Josh Engwer (TTU) Area Between Curves 22 January / 66
19 A Region that s both V-Simple & H-Simple Josh Engwer (TTU) Area Between Curves 22 January / 66
20 Area of a Region that s neither V-Simple nor H-Simple Josh Engwer (TTU) Area Between Curves 22 January / 66
21 Area of a Region that s neither V-Simple nor H-Simple Area(D) = Area(D 1 ) + Area(D 2 ) REMARK: Subdivide along a BP using a horizontal or vertical line. Josh Engwer (TTU) Area Between Curves 22 January / 66
22 Area of a V-Simple Region (Procedure) Proposition (Area of a V-Simple Region) Area(D) = largest x-value in D smallest x-value in D [ ] (Top BC) (Bottom BC) dx = b a [ ] g 2 (x) g 1 (x) dx Josh Engwer (TTU) Area Between Curves 22 January / 66
23 Area of a V-Simple Region (Procedure) Proposition (Area of a V-Simple Region) Area(D) = largest x-value in D smallest x-value in D [ ] (Top BC) (Bottom BC) dx = b a [ ] g 2 (x) g 1 (x) dx Josh Engwer (TTU) Area Between Curves 22 January / 66
24 Area of a V-Simple Region (Procedure) Proposition (Area of a V-Simple Region) Area(D) = largest x-value in D smallest x-value in D [ ] (Top BC) (Bottom BC) dx = b a [ ] g 2 (x) g 1 (x) dx Josh Engwer (TTU) Area Between Curves 22 January / 66
25 Area of a H-Simple Region (Procedure) Proposition (Area of a H-Simple Region) Area(D) = largest y-value in D smallest y-value in D [ ] (Right BC) (Left BC) dy = d c [ ] h 2 (y) h 1 (y) dy Josh Engwer (TTU) Area Between Curves 22 January / 66
26 Area of a H-Simple Region (Procedure) Proposition (Area of a H-Simple Region) Area(D) = largest y-value in D smallest y-value in D [ ] (Right BC) (Left BC) dy = d c [ ] h 2 (y) h 1 (y) dy Josh Engwer (TTU) Area Between Curves 22 January / 66
27 Area of a H-Simple Region (Procedure) Proposition (Area of a H-Simple Region) Area(D) = largest y-value in D smallest y-value in D [ ] (Right BC) (Left BC) dy = d c [ ] h 2 (y) h 1 (y) dy Josh Engwer (TTU) Area Between Curves 22 January / 66
28 Area Between Two Curves (Using V-Rects) EXAMPLE: Let region R be bounded by curves y = 2x, y = 4 2x 2, x = 2. Setup integral(s) to compute Area(R) using Vertical Rectangles (V-Rects). Josh Engwer (TTU) Area Between Curves 22 January / 66
29 Area Between Two Curves (Using V-Rects) EXAMPLE: Let region R be bounded by curves y = 2x, y = 4 2x 2, x = 2. Sketch & characterize region R (label BP s & BC s in terms of x) Notice subregions R 1, R 2 are each V-simple. Josh Engwer (TTU) Area Between Curves 22 January / 66
30 Area Between Two Curves (Using V-Rects) EXAMPLE: Let region R be bounded by curves y = 2x, y = 4 2x 2, x = 2. Sketch & characterize region R. (remove unnecessary clutter) Notice subregions R 1, R 2 are each V-simple. Josh Engwer (TTU) Area Between Curves 22 January / 66
31 Area Between Two Curves (Using V-Rects) Key Element: V-Rectangle (V-Rect) Josh Engwer (TTU) Area Between Curves 22 January / 66
32 Area Between Two Curves (Using V-Rects) k th V-Rect in R 1 : Width = x k Height = (Top BC) (Bottom BC) Area = (Height) (Width) Josh Engwer (TTU) Area Between Curves 22 January / 66
33 Area Between Two Curves (Using V-Rects) Width = [ x k k th V-Rect in R 1 : Height = 4 2 (xk )2] 2xk [ Area = 4 2xk 2 (x k )2] x k N [ Riemann Sum: Area(R 1 ) A N = 4 2xk 2 (xk ) 2] x k k=1 Josh Engwer (TTU) Area Between Curves 22 January / 66
34 Area Between Two Curves (Using V-Rects) Riemann Sum: Area(R 1 ) A N = N Integral: Area(R 1 ) = lim N A N = [ 4 2x k 2 (x k ) 2] x k k=1 largest x-coord. in R1 smallest x-coord. in R 1 ( 4 2x 2x 2 ) dx Josh Engwer (TTU) Area Between Curves 22 January / 66
35 Area Between Two Curves (Using V-Rects) Riemann Sum: Area(R 1 ) A N = N Integral: Area(R 1 ) = lim N A N = k=1 1 2 [ 4 2x k 2 (x k ) 2] x k ( 4 2x 2x 2 ) dx Josh Engwer (TTU) Area Between Curves 22 January / 66
36 Area Between Two Curves (Using V-Rects) k th V-Rect in R 2 : Width = x k Height = (Top BC) (Bottom BC) Area = (Height) (Width) Josh Engwer (TTU) Area Between Curves 22 January / 66
37 Area Between Two Curves (Using V-Rects) Width = x k [ k th V-Rect in R 2 : Height = 2xk 4 2 (xk )2] ] Area = [2 (xk )2 + 2xk 4 x k N [ ] Riemann Sum: Area(R 2 ) A N = 2 (xk ) 2 + 2xk 4 x k k=1 Josh Engwer (TTU) Area Between Curves 22 January / 66
38 Area Between Two Curves (Using V-Rects) Riemann Sum: Area(R 2 ) A N = N Integral: Area(R 2 ) = lim N A N = [ ] 2 (xk ) 2 + 2xk 4 x k k=1 largest x-coord. in R2 smallest x-coord. in R 2 ( 2x 2 + 2x 4 ) dx Josh Engwer (TTU) Area Between Curves 22 January / 66
39 Area Between Two Curves (Using V-Rects) Riemann Sum: Area(R 2 ) A N = N Integral: Area(R 2 ) = lim N A N = k=1 2 1 [ ] 2 (xk ) 2 + 2xk 4 x k ( 2x 2 + 2x 4 ) dx Josh Engwer (TTU) Area Between Curves 22 January / 66
40 Area Between Two Curves (Using V-Rects) Area(R) = Area(R 1 ) + Area(R 2 ) 1 ( = 4 2x 2x 2 ) 2 dx ( 2x 2 + 2x 4 ) dx = = 38 3 Josh Engwer (TTU) Area Between Curves 22 January / 66
41 Area Between Two Curves (Using H-Rects) EXAMPLE: Let region R be bounded by curves y = 2x, y = 4 2x 2, x = 2. Setup integral(s) to compute Area(R) using H-Rects. Sketch & characterize region R (label BP s & BC s in terms of y) Subdivide region (via dashed line) into four H-simple subregions R 3, R 4, R 5, R 6. Josh Engwer (TTU) Area Between Curves 22 January / 66
42 Area Between Two Curves (Using H-Rects) Key Element: H-Rectangle (H-Rect) Josh Engwer (TTU) Area Between Curves 22 January / 66
43 Area Between Two Curves (Using H-Rects) k th H-Rect in R 3 : Width = y k Length = (Right BC) (Left BC) Area = (Length) (Width) Josh Engwer (TTU) Area Between Curves 22 January / 66
44 Area Between Two Curves (Using H-Rects) Width = y k ) k th H-Rect in R 3 : Length = 1 2 y k ( y k ( ) 1 Area = 2 y k y k y k ( ) N Riemann Sum: Area(R 3 ) A N = 1 2 y k y k y k k=1 Josh Engwer (TTU) Area Between Curves 22 January / 66
45 Area Between Two Curves (Using H-Rects) Riemann Sum: Area(R 3 ) A N = N Integral: Area(R 3 ) = lim N A N = ( 1 2 y k + k=1 largest y-coord. in R3 smallest y-coord. in R 3 ) y k y k ( ) y + 12 y dy Josh Engwer (TTU) Area Between Curves 22 January / 66
46 Area Between Two Curves (Using H-Rects) Riemann Sum: Area(R 3 ) A N = N Integral: Area(R 3 ) = lim N A N = k=1 2 4 ( ( 1 2 y k y k ) 2 12 y 1 2 y + ) dy y k Josh Engwer (TTU) Area Between Curves 22 January / 66
47 Area Between Two Curves (Using H-Rects) k th H-Rect in R 4 : Width = y k Length = (Right BC) (Left BC) Area = (Length) (Width) Josh Engwer (TTU) Area Between Curves 22 January / 66
48 Area Between Two Curves (Using H-Rects) Width = y k ( ) k th H-Rect in R 4 : Length = 2 12 y k y k Area = y k y k N Riemann Sum: Area(R 4 ) A N = y k y k k=1 Josh Engwer (TTU) Area Between Curves 22 January / 66
49 Area Between Two Curves (Using H-Rects) N Riemann Sum: Area(R 4 ) A N = y k y k k=1 largest y-coord. in R4 Integral: Area(R 4 ) = lim N A N = smallest y-coord. in R 4 2 y dy Josh Engwer (TTU) Area Between Curves 22 January / 66
50 Area Between Two Curves (Using H-Rects) N Riemann Sum: Area(R 4 ) A N = y k y k k=1 Integral: Area(R 4 ) = lim N A N = y dy Josh Engwer (TTU) Area Between Curves 22 January / 66
51 Area Between Two Curves (Using H-Rects) k th H-Rect in R 5 : Width = y k Length = (Right BC) (Left BC) Area = (Length) (Width) Josh Engwer (TTU) Area Between Curves 22 January / 66
52 Area Between Two Curves (Using H-Rects) Width = y k k th H-Rect in R 5 : Length = y k ( ) Area = y k y k ( ) N Riemann Sum: Area(R 5 ) A N = y k y k k=1 Josh Engwer (TTU) Area Between Curves 22 January / 66
53 Area Between Two Curves (Using H-Rects) ( ) N Riemann Sum: Area(R 5 ) A N = y k y k k=1 ( ) 2 12 y Integral: Area(R 5 ) = lim N A N = largest y-coord. in R5 smallest y-coord. in R 5 2 dy Josh Engwer (TTU) Area Between Curves 22 January / 66
54 Area Between Two Curves (Using H-Rects) ( ) N Riemann Sum: Area(R 5 ) A N = y k y k k=1 ( ) 2 12 y Integral: Area(R 5 ) = lim N A N = dy Josh Engwer (TTU) Area Between Curves 22 January / 66
55 Area Between Two Curves (Using H-Rects) k th H-Rect in R 6 : Width = y k Length = (Right BC) (Left BC) Area = (Length) (Width) Josh Engwer (TTU) Area Between Curves 22 January / 66
56 Area Between Two Curves (Using H-Rects) k th H-Rect in R 6 : Width = y k Length = y k Area = ( ) y k yk Riemann Sum: Area(R 6 ) A N = N k=1 ( y k ) y k Josh Engwer (TTU) Area Between Curves 22 January / 66
57 Area Between Two Curves (Using H-Rects) Riemann Sum: Area(R 6 ) A N = N Integral: Area(R 6 ) = lim N A N = ( 2 1 ) 2 y k y k k=1 largest y-coord. in R6 smallest y-coord. in R 6 (2 12 y ) dy Josh Engwer (TTU) Area Between Curves 22 January / 66
58 Area Between Two Curves (Using H-Rects) ( ) N Riemann Sum: Area(R 6 ) A N = y k y k k=1 4 Integral: Area(R 6 ) = lim N A N = (2 12 ) y dy 2 Josh Engwer (TTU) Area Between Curves 22 January / 66
59 Area Between Two Curves (Using H-Rects) Area(R) = Area(R 3 ) + Area(R 4 ) + Area(R 5 ) + Area(R 6 ) ( ) 2 1 = y y dy ( ) y dy y dy (2 12 ) y dy = = Josh Engwer (TTU) Area Between Curves 22 January / 66
60 The Facts of Life (according to WeBWorK)...and then WeBWorK decreed: The Good News: Many HW problems just want the integral(s) setup. The Bad News: Many HW problems want the integral(s) computed. So let s briefly recap computation of basic integrals from Calculus I... Josh Engwer (TTU) Area Between Curves 22 January / 66
61 Indefinite Integral Rules (from Calculus I) Here, C R is called the constant of integration. Also, k R. Zero Rule: 0 dx = C Constant Rule: k dx = kx + C Constant Multiple Rule: kf (x) dx = k f (x) dx Sum/Diff Rule: [f (x) ± g(x)] dx = f (x) dx ± g(x) dx Power Rule: x n dx = 1 ( ) n + 1 xn+1 + C provided n R \ { 1} e x dx = e x + C ( ) a x dx = ax ln a + C provided a R + \ {1} 1 dx = ln x + C x NOTATION: R \ { 1} means All real numbers except 1 NOTATION: R + \ {1} means All positive real numbers except 1 Josh Engwer (TTU) Area Between Curves 22 January / 66
62 Indefinite Integral Rules (from Calculus I) Here, C R is called the constant of integration. sin x dx = cos x + C cos x dx = sin x + C sec 2 x dx = tan x + C sec x tan x dx = sec x + C csc 2 x dx = cot x + C csc x cot x dx = csc x + C 1 dx = arcsin x + C 1 x 2 1 dx = arctan x + C 1 + x2 1 x dx = arcsec x + C x 2 1 Josh Engwer (TTU) Area Between Curves 22 January / 66
63 Fundamental Theorem of Calculus (FTC) Theorem Let function f C 1 [a, b]. Then 1 x=1 b a 2 f (x) dx = f (b) f (a) ( WORKED EXAMPLE: Compute I = x 4 3 ) dx. 1 2 ( x 4 3 ) [ ] x=2 [ ] [ ] 1 dx = 5 x5 FTC 1 1 3x = 5 (2)5 3(2) 5 (1)5 3(1) = 16 5 WORKED EXAMPLE: Compute I = 3π/4 π/6 3π/4 π/6 sin θ dθ. [ ] θ=3π/4 FTC sin θ dθ = cos θ = [ cos(3π/4)] [ cos(π/6)] θ=π/6 ( ) ( ) = + = Josh Engwer (TTU) Area Between Curves 22 January / 66
64 Change of Variables (u-substitution) WORKED EXAMPLE: Evaluate I = xe x2 CV: Let u = x 2, then du = 2x dx = x dx = 1 2 du ( ) = xe x2 dx CV 1 = e u 2 du = 1 2 eu + C CV = 1 2 ex2 + C WORKED EXAMPLE: Evaluate I = 3 xe x2 2 dx. dx. CV: Let u = x 2, then du = 2x dx = x dx = 1 2 du and u( 2) = ( 2) 2 = 4 and u(3) = (3) 2 = ( ) [ = xe x2 dx CV 1 1 = e u 2 du = 2 eu 2 4 ] u=9 u=4 FTC = 1 ( e 9 e 4) 2 Josh Engwer (TTU) Area Between Curves 22 January / 66
65 Nonelementary Integrals Definition A nonelementary integral is an integral whose antiderivative cannot be expressed in a finite closed form. Here s a small list of nonelementary integrals (there are many, many more): e x xe e x2 dx x dx x dx sin(x 2 ) dx cos(e x ) dx e cos x dx 1 x + x4 dx ln(ln x) dx e x 1 dx 1 sin x ln x dx dx sin(sin x) dx x 1 x x dx x x dx arctan(ln x) dx When computing integrals, avoid nonelementary integrals! If using V-Rects leads to a nonelementary integral, use H-Rects instead. If using H-Rects leads to a nonelementary integral, use V-Rects instead. Josh Engwer (TTU) Area Between Curves 22 January / 66
66 Fin Fin. Josh Engwer (TTU) Area Between Curves 22 January / 66
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Continuous Random Variables: pdf s
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