Chapter 5

5.3 Definite IntegralsObj: find area using definite integrals

Review

Estimate the area using 4 subintervals between f(x)=x3 and the x-axis from 0 to 2 using left and right sums.

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Riemann Sum

• Riemann Sum

• Left Sum• Right Sum• Midpoint Sum• Trapezoidal Sum

1

( )n

i ii

f c x

Definite Integral

If f is defined on the closed interval [a, b] and the limit exists, then

If f is continuous on [a, b], then f is integrable on [a, b].

0 1

lim ( ) ( )n b

i i ax i

f c x f x dx

Area

IF f is continuous and nonnegative on the closed interval [a, b], then the area bounded by f, the x-axis and the vertical lines a and b is

Area = ( )b

af x dx

Examples

2 2

24 x dx

5

0( 2)x dx

Rules1. If f is defined at x = a, .

2. If f is integrable on [a, b], then

3. If c is on [a, b], then

( ) 0a

af x dx

( ) ( )a b

b af x dx f x dx

( ) ( ) ( )c b b

a c af x dx f x dx f x dx

Assignment

• page 314 15 – 42 by 3, 47, 48