Review5.1-5.5 Review5.1-5.5

5.1 Do you know RAM theory?

Which over- and under-approximates increasing and decreasing functions?How to compute given an equation?How to compute given data?

( ) when f(x) 0b

a

Area f x dx

( ) when f(x) 0b

a

Area f x dx

( ) = (area above x-axis) - (area below x-axis)b

a

f x dx

Net Area:

5.2 The Definite Integral

Integrals on the TI-89

Syntax: To evaluate …

( ) b

a

f x dx

2nd 7 (f(x),x,a,b)

5.3 Properties of Definite Integrals:

( ) ( ) a b

b a

f x dx f x dx 1. Order of Integration

( ) = 0a

a

f x dx2. Zero

( ) ( ) b b

a a

k f x dx k f x dx 3. Constant Multiple

[ ( ) ( )] ( ) ( ) b b b

a a a

f x g x dx f x dx g x dx4. Sum and Difference

( ) + ( ) ( ) b c c

a b a

f x dx f x dx f x dx 5. Additivity

5.3 MVT for Integrals…

Somewhere between an inscribed and circumscribed rectangle is a rectangle whose area is exactly the area of the region under the curve.

( ) = ( )( )b

a

f x dx f c b a

5.3 Average Value Theorem

The value f(c) from the MVT is the average value of f on [a,b]…

1( ) ( )

( )

b

a

f c f x dxb a

5.4 Fund. Thm of Calc: Part I

( ) =F(b)-F(a)b

a

f x dx

5.4 Fund Thm of Calc: Part 2

[ ( ) ] =f(x)x

a

df t dt

dx

5.5 Trapezoidal Rule

0 1 2 1( ) ( ) 2 ( ) 2 ( ) 2 ( ) ( )

2

b

n na

b af x dx f x f x f x f x f x

n

where n is the number of subintervals.

The trapezoidal rule overapproximates CCU functions and underapproximates CCD functions.