Chapter 5-1 Notes
b2
r w h w h
A=lw
b1 l b
A=
bh A= pr2 A=
h(b1+b2)
(a) By reading values from
the given graph of f, use five rectangles to find a lower estimate
and an upper estimate for the area under the given graph of f from x
= 0 to x = 10. In each case sketch the rectangles that you use. Use these equations: Rn=Dx[ f(x1) + f(x2) +…+
f(xn-1)] Ln=Dx[ f(x0) + f(x1) +…+
f(xn-1)]
Ex 1)
10 5 0 5 y = f(x)
|
|
X0 |
X1 |
X2 |
X3 |
X4 |
X5 |
|
X |
0 |
2 |
4 |
6 |
8 |
10 |
|
Y |
0.5 |
1.5 |
3 |
4 |
4.5 |
5 |
(a)
Ex
2) Find an expression for the area under the graph of f as a limit. Do
not evaluate the limit.
F(x) = 
, 
Dx = 
Ex
3) The speed of a runner increased steadily during the first three seconds of a
race. Her speed at half-second intervals is given in the table.
Find lower and upper estimates for
the distance that she traveled during these three seconds.
|
t(s) |
0 |
0.5 |
1.0 |
1.5 |
2.0 |
2.5 |
3.0 |
|
v(ft/s) |
0 |
6.2 |
10.8 |
14.9 |
18.1 |
19.4 |
20.2 |
R6
= .5(6.2+10.8+14.9+18.1+19.4+20.2) = 44.8
L6
= .5(0+6.2+10.8+14.9+18.1+19.4) = 34.7
Chapter 5-2 Notes
Ex
1)
a) Find the Riemann sum for f(x)
= x- 2sin2x, 0
x
3, with six terms, taking the sample points to be right
endpoints. (Give your answer correct to six decimal places.)
.5(1.182942+.818595+1.21776+3.513605+4.417849+3.558831)
=7.354791
b) Repeat part a) with midpoints as the sample
points.
.5(.708851+1.24499+.053056+2.451567+4.205060+4.161081)
=6.412303
Ex
2)
Use
the form of the definition of the integral to evaluate the integral.
The first thing to do is to change this to limit form. Let’s assume that we are using right endpoints.