Calc05_2 - 5.2 Definite Integrals Greg Kelly Hanford High...

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5.2 Definite Integrals Greg Kelly, Hanford High School, Richland, Washington
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When we find the area under a curve by adding rectangles, the answer is called a Rieman sum . 2 1 1 8 V t = + subinterval partition The width of a rectangle is called a subinterval . The entire interval is called the partition . Subintervals do not all have to be the same size.
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2 1 1 8 V t = + subinterval partition If the partition is denoted by P , then the length of the longest subinterval is called the norm of P and is denoted by . P As gets smaller, the approximation for the area gets better. P ( 29 0 1 Area lim n k k P k f c x = = if P is a partition of the interval [ ] , a b
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( 29 0 1 lim n k k P k f c x = is called the definite integral of over . f [ ] , a b If we use subintervals of equal length, then the length of a subinterval is: b a x n - = The definite integral is then given by: ( 29 1 lim n k n k f c x →∞ =
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( 29 1 lim n k n k f c x →∞ = Leibnitz introduced a simpler notation for the definite integral: ( 29 ( 29 1 lim n b k a n k f c x f x dx →∞ = = Note that the very small change in x becomes dx .
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