Download presentation

Presentation is loading. Please wait.

Published byWill Stalker Modified over 2 years ago

1
Areas, Volumes, Work Sections 5.1, 5.2, 5.3, 5.4 Chapter 5. Applications of Integration

2
5.1 Area between curves

3
5.1 Area between two curves Two curves y = f(x), y = g(x) g(x) ≤ f(x), a ≤ x ≤ b R = { (x,y) | a ≤x ≤ b, g(x) ≤ y ≤ f(x) } x y a y = f(x) b x (x,y) What is the area of R ? First, assume f(x) ≥ 0 and g(x) ≥ 0 on [a,b] y = g(x)

4
5.1 Area between two curves Two curves y=f(x), y=g(x) g(x) ≤ f(x), a ≤ x ≤ b R = { (x,y) | a ≤x ≤ b, g(x) ≤ y ≤ f(x) } x y a y = f(x) b First, assume f(x) ≥ 0 and g(x) ≥ 0 on [a,b] y = g(x) Af Area = Af – Ag = Ag Af - Ag y = g(x)

5
General Case Two curves y=f(x), y=g(x) g(x) ≤ f(x), a ≤ x ≤ b R = { (x,y) | a ≤x ≤ b, g(x) ≤ y ≤ f(x) } x y a y = f(x) b y = g(x) Idea: shift up using transformations y = f(x)+ K and y = g(x) + K Shift does not change the areas

6
5.1 Area between two curves x y a y = f(x)+K b y = g(x) y = g(x) + K

7
Area between two curves x y a y = f(x) b y = g(x)

8
Intersections of graphs Often, a or b or both correspond to points where graphs y = f(x) and y = g(x) intersect To find intersection points, solve equation f(x) = g(x)

9
5.2 Volumes

10
Volumes of simple 3D objects L W hh r V = (L W) hV = (π r 2 ) h In both cases, V = A h = (Area of the base) (height)

11
Generalized Cylinder V = A h = (Area of the base) (height) h A

12
General 3D shape Cross section

13
Area of a cross section x x A(x)

14
Slicing =x n x1x1 x i-1 xixi x a b x0=x0=

15
i th slice x i-1 xixi x

16
i th slice – approximation by cylinder x i-1 xixi x

17
i th slice – approximation by cylinder x i-1 xixi x A(x i ) ΔxΔx V i ≈ A(x i ) Δx V(i th slice) ≈ V(i th cylinder) = Area of the base height = A(x i ) Δx ViVi

18
Total Volume Total volume = sum of volumes of all slices ≈ sum of volumes of all approximating cylinders

19
Volume as integral of areas of cross sections x x A(x) b a x0=x0=

20
Solids of revolution

21
Rotate a plane region around a line – axis of rotation

23
Volumes of solids of revolution using “washers”

24
“Washer” – region between two concentric circles r in r out Area of the “washer” = = A(outer disk) – A(inner disk) =π (r out ) 2 -π (r in ) 2 = π [ (r out ) 2 - (r in ) 2 ]

25
Cross sections are washers

26
Area of cross section x x b a r in (x) r out (x) A(x) = π [r 2 out (x) - r 2 in (x) ]

27
Volume x x b a r in (x) r out (x) A(x) = π [r 2 out (x) - r 2 in (x) ]

28
5.3 Volumes by cylindrical shells

29
Cylindrical Shell

30
Volume of Cylindrical Shell h R out R in V = V(outer cyllinder) – V(inner cyllinder)= = (π R out 2 ) h - (π R in 2 ) h = π(R out 2 - R in 2 ) h= = π(R out - R in ) (R out + R in ) h Let Δr = thickness = R out – R in Let r = average radius = (R out + R in )/2 V = (2 π r) h Δr

31
Volume of Cylindrical Shell V = (2 π r) h Δr = = circumference height thickness h r ΔrΔr

32
Solid of revolution

34
Approximate the region using rectangles

35
Rotating the rectangles, we get cylindrical shells =x n x1x1 x i-1 xixi x a b x0=x0= y h(x i ) ΔxΔx ΔxΔx r(x i )

36
Rotating the rectangles, we get cylindrical shells =x n x1x1 x i-1 xixi x a b x0=x0= y h(x i ) ΔxΔx ΔxΔx r(x i )

37
Volume of i-th shell =x n x1x1 x i-1 xixi x a b x0=x0= y h(x i ) ΔxΔx ΔxΔx r(x i ) V i = circumference height thickness = = 2 π r(x i ) h(x i )Δx

38
Approximation by shells V i = 2 π r(x i ) h(x i ) Δx

39
Volume by shells x x a b y h(x) r(x)

40
5.4 Work

41
Work done by constant force Work = Force Displacement

Similar presentations

OK

Chapter 6 – Applications of Integration 6.3 Volumes by Cylindrical Shells 1Erickson.

Chapter 6 – Applications of Integration 6.3 Volumes by Cylindrical Shells 1Erickson.

© 2018 SlidePlayer.com Inc.

All rights reserved.

Ads by Google

Ppt on natural numbers Ppt on earth dam spillway Ppt on formal education examples Ppt on bond length trend Ppt on internal trade for class 11th Ppt on area of rectangle and square Ppt on mobile computing from iit bombay Ppt on ancient 7 wonders of the world Download ppt on mind controlled robotic arms building Ppt on regional trade agreements signed