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Aim: Finding Area Course: Calculus
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Aim: An introduction to the 2nd central Idea of Calculus.
5Evaluate: dx
x
3
2
cos 4
cos
xdx
x
Aim: Finding Area Course: Calculus
Area
A = bh
b
h
Rectangle
A = 1/2 bh
b
h
Triangle
Aim: Finding Area Course: Calculus
B
A CD
mABD = 60.02
B
A CD
Area
Inscribed or circumscribed polygons
(circle) = lim ( )nnA A P
Aim: Finding Area Course: Calculus
Life Gets Complex
5
4
3
2
1
1 2
Lake Wallawalla
Aim: Finding Area Course: Calculus
Life Gets Complex
5
4
3
2
1
1 2
Lake Wallawalla
Aim: Finding Area Course: Calculus
5
4
3
2
1
1 2
Approximating the Area of a Plane Region
How would we approximate the area of the shaded region under the curve f(x) =
-x2 + 5 in the interval [0, 2]?
height – value of f(x)
Arect = xf(x)
Arect = w·l
Approximate area is sum of areas of 2 rectangles having equal widths
inscribed rectangles
f(1)
f(2)
1
2
width – value of x
A (1)f(1) + (1)f(2)
Aunder curve A1 + A2
= (1)[-(1)2 + 5] + (1)[-(2)2 + 5]
A 4 + 1 = 5
Aim: Finding Area Course: Calculus
5
4
3
2
1
1 2
Approximating the Area of a Plane Region
How would we approximate the area of the shaded region under the curve f(x) =
-x2 + 5 in the interval [0, 2]?
2/5 4/5 6/5 8/5 2
5 intervals of equal width -
2/5 units
f(2/5)f(4/5)
f(6/5)
f(8/5)
f(2)
height of f(x)
A = Δxf(x)
A = w·l
right endpoints are length of each rectangle
right endpoints
inscribed rectangles 1 2
3 45
l of rectangle 1 = f(2/5) = -(2/5)2 + 5 = 4.84
w of rectangle 1 = 2/5
Arect 1 = (2/5)4.84 = 1.936
Aim: Finding Area Course: Calculus
5
4
3
2
1
1 2
Approximating the Area of a Plane Region
2/5 4/5 6/5 8/5 2
f(2/5)f(4/5)
f(6/5)
f(8/5)
f(2)
right endpoints are length of each
rectangle
right endpoints
inscribed rectangles 1 2
3 45
l2 = f(4/5) = -(4/5)2 + 5 = 4.36
l3 = f(6/5) = -(6/5)2 + 5 = 3.56
l4 = f(8/5) = -(8/5)2 + 5 = 2.44
l5 = f(2) = -(2)2 + 5 = 1
l 1 = f(2/5) = -(2/5)2 + 5 = 4.84
Arect 1 = 1.936
Arect 2 = 1.744
Arect 3 = 1.424
Arect 4 = .976
Arect 5 = .4
Sum of 5 areas = 6.48
Aim: Finding Area Course: Calculus
Approximating the Area of a Plane Region
How would we approximate the area of the shaded region under the curve f(x) =
-x2 + 5 in the interval [0, 2]?
5 intervals of equal width -
2/5 units
height of f(x)
A = Δxf(x)
A = w·l
2/5 4/5 6/5 8/5 2
5
4
3
2
1
1 2
f(4/5)
f(6/5)
f(2/5)f(0)
f(8/5)
left endpoints
circum-scribed
rectangle
left endpoints are length of each rectanglel of rectangle 1 = f(0) = -(0)2 + 5 = 5
w of rectangle 1 = 2/5
Arect 1 = (2/5)5 = 2
1 23 4
5
Aim: Finding Area Course: Calculus
Approximating the Area of a Plane Region
2/5 4/5 6/5 8/5 2
5
4
3
2
1
1 2
f(4/5)
f(6/5)
f(2/5)f(0)
f(8/5)
left endpoints
circum-scribed
rectangle
left endpoints are length of each
rectangle
l 1 = f(0) = -(0)2 + 5 = 5
Arect 1 = 2
l2 = f(2/5) = -(2/5)2 + 5 = 4.84
l3 = f(4/5) = -(4/5)2 + 5 = 4.36
l4 = f(6/5) = -(6/5)2 + 5 = 3.56
l5 = f(8/5) = -(8/5)2 + 5 = 2.44
Arect 2 = 1.936
Arect 3 = 1.744
Arect 4 = 1.424
Arect 5 = .976
Sum of 5 areas = 8.08
Aim: Finding Area Course: Calculus
Approximating the Area of a Plane Region
How would we approximate the area of the shaded region under the curve f(x) =
-x2 + 5 in the interval [0, 2]?
Approximate area is sum of areas of rectangles
2/5 4/5 6/5 8/5 2
5
4
3
2
1
1 2
f(4/5)
f(6/5)
f(2/5)f(0)
f(8/5)
left endpoints
6.48 < area of region < 8.08
2/5 4/5 6/5 8/5 2
5
4
3
2
1
1 2
f(2/5)f(4/5)
f(6/5)
f(8/5)
f(2)
right endpoints
increasing number of rectangles – closer approximations.
lower sum upper sum
Aim: Finding Area Course: Calculus
Devising a Formula
•Let a be left endpoint of the interval of area to be found
•Let b be right endpoint of interval of area4
3.5
3
2.5
2
1.5
1
0.5
1 2
f x = x2
lower sumpartition into n intervals
xa b
•height of rectangle 1 is y0
12
yn - 1yn - 2
•height of rectangle 2 is y1
•height of rectangle 3 is y2
etc.•height of last rectangle is
yn - 1
y0
y1
yn - 1
b ax
n
width
Aim: Finding Area Course: Calculus
Devising a Formula
•Using left endpoint to approximate area under the curve is
0 1 2 1n
b ay y y y
n
4
3.5
3
2.5
2
1.5
1
0.5
1 2
f x = x2
lower sum
xa b
12
yn - 1yn - 2y0
y1
yn - 1
the more rectangles the better the
approximation
the exact area?
take it to the limit!
0 1 2 1lim nn
b ay y y y
n
left endpoint formula
Aim: Finding Area Course: Calculus
•Using right endpoint to approximate area under the curve is
Right Endpoint Formula
4
3.5
3
2.5
2
1.5
1
0.5
1 2
f x = x2
x
upper sum
a by0
y1
yn - 1
yn
1 2 3 n
b ay y y y
n
1 2 3lim nn
b ay y y y
n
right endpoint formula
1 3 5 2 1
2 2 2 2
lim nn
b ay y y y
n
midpoint formula
y2
Aim: Finding Area Course: Calculus
30
25
20
15
10
5
1 2 3
Model Problem
Approximate the area under the curve y = x3 from x = 2 to x = 3 using four left-endpoint rectangles.
b ax
n
width:
3 2 1
4 4x
2 39
4
5
2
11
4
3(2) 2 8f height: y0 =
y0
y1
y2
y3
0 1 2 1n
b ay y y y
n
39 9
( )4 4
f
y1 =
35 5
( )2 2
f
y2 =
311 11
( )4 4
f
y3 =
Aim: Finding Area Course: Calculus
30
25
20
15
10
5
1 2 3
Model Problem
Approximate the area under the curve y = x3 from x = 2 to x = 3 using four left-endpoint rectangles.
2 39
4
5
2
11
4
y0
y1
y2
y3
0 1 2 1n
b ay y y y
n
3 3 3
31 9 5 112
4 4 2 4A
89313.953
64A
Aim: Finding Area Course: Calculus
30
25
20
15
10
5
1 2 3
Model Problem
Approximate the area under the curve y = x3 from x = 2 to x = 3 using four right-endpoint rectangles.
2 39
4
5
2
11
4
y1
y2
y3
1 2 3 n
b ay y y y
n
3 3 3
31 9 5 113
4 4 2 4A
119718.703
64A
y4
Aim: Finding Area Course: Calculus
Homework
Approximate the area under the curve y = x3 from x = 2 to x = 3 using four midpoint rectangles.
Aim: Finding Area Course: Calculus
Model Problem
Approximate the area under the curve y = 6 + 2x - x3 for [0, 2] using 8 left endpoint rectangles. Sketch the graph and regions.
Aim: Finding Area Course: Calculus
Model Problem
Approximate the area under the curve y = 4 – x2 for [-1, 1] using 4 inscribed rectangles.
