4.2 Area
...
Origins of Calculus
Day 2
‐ two classic
problems:
Q
the tangent line problem
the P
area problem
a
b
The Area of a Plane Region
.
Def
Area = b .h
rectangle
1
Example: Use 5 rectangles (equal width) to "squeeze" the exact area between two approximations of the area of the region lying between the graph of f (x) = x 2 and the x­axis between x = 0 and x = 5.
width =
intervals:
total length
# rectangles
=
5 ­ 0
5
=1
[0, 1] , [1, 2] , [2, 3] , [3, 4] , [4, 5]
0
5
2
left endpoints:
|
|
|
|
0
1 2 3 4
0
1 2 3 4
|
|
|
5
|
5
3
right endpoints:
|
|
|
|
0
1 2 3 4
0
1 2 3 4
|
|
|
5
|
5
4
Theory and Terminology:
(a, f (a))
(b, f (b))
y = f (x)
a
b
5
[a, b] into n subintervals
subdivide
(a, f (a))
(b, f (b))
y = f (x)
a ( x1 x2 x3 . . . xi . . . xn­1 b
( x0
(xn(
width = x =
total length
# rectangles
= b ­ a
n
6
(a, f (a))
(b, f (b))
y = f (x)
a( x1 x2 x3 . . . xi . . . x
n­1
}
}
}
( x0 x x x
b
(xn(
Notice:
x0 = a
.
x1 = a + 1 x
.
x2 = a + 2 x
.
x3 = a + 3 x
. . .
xi = a + i . x
. . .
.
xn = a + n x = b
7
i th
subinterval
3.1
{
f (Mi)
Ex
Va
lu
tre
eT
}
me
hm
.
f (mi)
8
Defs.
f (mi) . x
circumscribed rectangle = f (Mi) . x
Area of the i th inscribed rectangle = Area of the i th
Lower sum = s(n) = n
f (mi) . x
total area
of inscribed
rectangles
f (Mi) . x
total area
of circumscribed
rectangles
i = 1
Upper sum = S(n) = n
i = 1
s(n) Area of
Region
S(n) 9
Thm 4.3
Limit of the Lower & Upper Sums
Let f be continuous and nonnegative on the interval [a, b].
s(n) = lim
lim
n
n
= lim
n
= lim
n
n
f (mi) . x
i = 1
n
f (Mi) . x
i = 1
S(n) 10
i th
subinterval
}
f (ci)
xi ­ 1
ci
xi
Area Under the Curve = Area of the Rectangle
11
Def. of the
Area of a Region
in the Plane
Let f be continuous and nonnegative on [a, b].
The area of the region bounded by the graph of f ,
the x­axis, and the vertical lines x = a and x = b is
Area = lim
n
where
n
f (ci) . x
i = 1
x = b ­ a
n
12
Find the Area (using the limit process) of the region between the graph of f (x) = x2 and the x­axis on [0, 5].
Example:
0
5
13
Assignment
p262-263 #23-29 odd,
#47, 49, 51, 53,
#57, 59, 72
limit process
horizontal rectangles
(see Ex. 7 p261)
14
p263 #60
Use the limit process to find the area of the region between the graph of the function and the y­axis over the indicated y­interval. Sketch the region.
f ( y) = 4y ­ y2 , 1 y 2
y
2
_
1
x
15
16