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Chapter 5
5.3 Definite IntegralsObj: find area using definite integrals
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Review
Estimate the area using 4 subintervals between f(x)=x3 and the x-axis from 0 to 2 using left and right sums.
dxxxx )31264( 23
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Riemann Sum
• Riemann Sum
• Left Sum• Right Sum• Midpoint Sum• Trapezoidal Sum
1
( )n
i ii
f c x
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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
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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
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Examples
2 2
24 x dx
5
0( 2)x dx
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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
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Assignment
• page 314 15 – 42 by 3, 47, 48