4.3 Reimann Sums
and Definite Integrals
Correction to Assignment Sheet:
you do NOT need to do 45, 46
(please do 47, 48)
1
The Definite Integral is the Area of a Region
If f is continuous and nonnegative on the closed interval [a, b], then the area of the region bounded by the graph of f, the x­axis, and the vertical line x = a and x = b is given by
lim
Δxi->0
∞
Ʃ
f(xi) Δxi
height
base
i=1
2
Set up a definite integral that yields the area of the region. (Do not evaluate the integral.)
3
Set up a definite integral that yields the area of the region. (Do not evaluate the integral.)
4
Sketch the region corresponding to each definite integral. Then evaluate each integral using a geometric formula.
5
Sketch the region corresponding to each definite integral. Then evaluate each integral using a geometric formula.
6
Evaluate
7
Properties of Definite Integrals
1.
2.
8
Definitions of Two Special Definite Integrals
9
Evaluate the integrals using the following values:
3
∫
3
∫
x2dx = 15
1
∫
∫
xdx = 4
1
3
1.
3
5xdx =
∫
1
3
2.
3
2
x dx =
3
1
dx = 2
3
∫
(x2 + 7)dx =
1
1
4.
∫
2x2dx =
3
10
Additive Interval Property
If f is integrable on the three closed intervals determined by a, b, and c, then 11
/
12
The graph of f consists of line segments, as shown in the figure. Evaluate each definite integral by using geometric formulas.
13
Calc In Motion - Definite Integration :)
14
The graph of f consists of a semicircle and line segments, as shown in the figure. Evaluate each definite integral by using geometric formulas.
a.
b.
c.
d.
e.
15
16