Integration Review Part I When you see the words… This is what you think of doing… A Riemann Sum equivalent to the definite integral is… - () b a f x dx 1 0 lim ( ) for a x b x f x x 1
Integration Review Part I
When you see the words… This is what you think of doing…
A Riemann Sum equivalent to the definite integral
is…
-
( )b
af x dx
10lim ( ) f or a x bx
f x x
1
Integration Review Part I
When you see the words… This is what you think of doing…
If f is continuous and bounded on an interval containing x = a and
F’(x) = f(x) for all x in the interval
( ) ( ) , then...x
aF x f t dt
2
Integration Review Part I
When you see the words… This is what you think of doing…
F (b) – F(a) where F’ = f
( )b
af x dx
3
Integration Review Part I
When you see the words… This is what you think of doing…
-( )a
bf x dx
4
( )b
af x dx
Integration Review Part I
When you see the words… This is what you think of doing…
-
5
( )c
af x dx
( ) ( )b c
a bf x dx f x dx
Integration Review Part I
When you see the words… This is what you think of doing…
f(x)
6
( )x
a
d f t dtdx
Integration Review Part I
When you see the words… This is what you think of doing…
f( g(x) ) * g’(x)
7
( ) ( )g x
a
d f t dtdx
Integration Review Part I
When you see the words… This is what you think of doing…
0
8
( )a
af x dx
Integration Review Part I
When you see the words… This is what you think of doing…
Trapezoid Rule -
9
( )b
af x dx
0 1 2 1( 2 2 ... 2 )2 n nh y y y y y
Integration Review Part I
When you see the words… This is what you think of doing…
If f is increasing on the interval [a,b], thena Left Riemann Sum (overestimates, underestimates) the true value of
Underestimates
10
( )b
af x dx
Integration Review Part I
When you see the words… This is what you think of doing…
If f is decreasing on the interval [a,b], thena Left Riemann Sum (overestimates, underestimates) the true value of
Overestimates
11
( )b
af x dx
Integration Review Part I
When you see the words… This is what you think of doing…
If f is increasing on the interval [a,b], thena Right Riemann Sum (overestimates, underestimates) the true value of
Overestimates
12
( )b
af x dx
Integration Review Part I
When you see the words… This is what you think of doing…
If f is decreasing on the interval [a,b], thena Right Riemann Sum (overestimates, underestimates) the true value of
Underestimates
13
( )b
af x dx
Integration Review Part I
When you see the words… This is what you think of doing…
If f is concave upward on the interval [a,b], then the Midpoint Riemann Sum (overestimates, underestimates) the true value of
Underestimates
14
( )b
af x dx
Integration Review Part I
When you see the words… This is what you think of doing…
If f is concave downward on the interval [a,b], then the Midpoint Riemann Sum (overestimates, underestimates) the true value of
Overestimates
15
( )b
af x dx
Integration Review Part I
When you see the words… This is what you think of doing…
If f is concave downward on the interval [a,b], then the Trapezoid Rule Sum (overestimates, underestimates) the true value of
Underestimates
16
( )b
af x dx
Integration Review Part I
When you see the words… This is what you think of doing…
If f is concave upward on the interval [a,b], then the Trapezoid Rule Sum (overestimates, underestimates) the true value of
Overestimates
17
( )b
af x dx
Integration Review Part I
When you see the words… This is what you think of doing…
Displacement (Integral)
.
18
( )b
av t dt
Integration Review Part I
When you see the words… This is what you think of doing…
.
19
nu du1
1nu C
n
Integration Review Part I
When you see the words… This is what you think of doing…
.
20
ue du ue C
Integration Review Part I
When you see the words… This is what you think of doing…
.
21
ue du ue C
Integration Review Part I
When you see the words… This is what you think of doing…
.
22
duu ln| |u C