pg 259-4.2-Area
-Use sigma notation to write and evaluate a sum.
Sigma Notation or Summation NotationThe sum of n terms a1, a2, a3, ...., an is written as
where i is the index of summation, ai is the ith
term of the sum, and the upper and lower bounds
of the summation are n and 1.
Ex. 1-Find the sum
Theorem 4.2-Summation Formulas
1)
2)
3)
Ex 2-Evaluate
10,000
4)
for n = 10, 100, 1000,
-Understand the concept of area
Read pg 261
-Approximate the area of a plane region
Ex. 3-Use eight rectangles in the figure to the
find the two approximations of the area of the
region lying between the graph of
and the x-axis between x = 0 and x = 2.
Read pg 263-Subdivide into equal parts
width (b-a)/n
Inscribed rectangles is when the rectangles lay
inside the region (Lower sum)
Circumscribed rectangles is when the
rectangles lay outside the region (Upper sum)
Lower Sum
Upper Sum
s(n) ≤ Area of region ≤ S(n)
Ex. 4-Find the upper and lower sums for the
region bounded by the graph of
and the x-axis between x = 0 and x = 3.
Thm 4.3-Limits of the Lower and Upper SumsLet f be continuous and nonnegative on the
interval [a, b]. The limit as n→∞ of both the
lower and upper sums exist and are equal to
each other. That is
where
and f(mi) and f(Mi) are the
minimum and maximum values of f on the
subinterval.
Definition of the Area of a Region in the PlaneLet f be continuous and nonnegative on the
interval [a, b]. The are of the region bounded by
the graph of f, the x-axis, and the vertical lines
x = a and x = b is
Area =
where
Ex. 5-Find the area of the region bounded by the
graph
the x-axis, and the vertical
lines x = 0 and x = 1.
Ex. 6-Find the area of the region bounded by the
graph of
the x-axis and the vertical
lines x = 1 and x = 3.
Ex. 7-Find the area of the region bounded by the
graph of
and the y-axis for 1 ≤ y ≤ 2.