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

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

5.1 Area between curves

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)

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)

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

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

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

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)

5.2 Volumes

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)

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

General 3D shape Cross section

Area of a cross section x x A(x)

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

i th slice x i-1 xixi x

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

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

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

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

Solids of revolution

Rotate a plane region around a line – axis of rotation

Volumes of solids of revolution using “washers”

“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 ]

Cross sections are washers

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) ]

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

5.3 Volumes by cylindrical shells

Cylindrical Shell

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

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

Solid of revolution

Approximate the region using rectangles

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 )

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 )

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

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

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

5.4 Work

Work done by constant force Work = Force  Displacement

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