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03Integration with a Boundary Condition and The Definite Integral2

03Integration with a Boundary Condition and The Definite Integral2

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I NTEGRATION WITH A B OUNDARY C ONDITION AND T HE D EFINITE I NTEGRAL Initial Value Problems When we integrate a function ) ( x f we get a family of functions C x F + ) ( , but sometimes we are interested in one particular function. We can identify this function by specifying a specific point it passes through. Ex . 2 2 3 ) ( x x x f - + = 1 ) 0 ( = F This question can also be presented as 2 2 3 x x dx dy - + = 1 ) 0 ( = y Where the unknown is not a number, but a function. This type of problem (involving the derivative of an unknown function) is called a differential equation. The Definite Integral Area Interpretation positive area 1 A 3 A a 2 A b negative area Total Area = Area over x -axis – Area under x -axis = 2 3 1 A A A - +
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Notation : b a dx x f ) ( is the definite integral of f from a to b a is the lower limit, b is the upper limit, ) ( x f is the integrand Ex . Sketch and evaluate using geometric formulae. 1) 4 1 2 dx 2) - + 2 1 ) 2 ( dx x 3) - 2 0 ) 1 ( dx x Note: Must split into two integrals. (Total area is 0). ) ( ) ( x f x A = A 0 ) ( = a A = b a dx x f A ) ( A b A = ) ( a b ) ( ) ( x f x A = let C x A x F + = ) ( ) ( Area = [ ] [ ] A A a A b A c a A c b A a F b F = - = - = + - + = - 0 ) ( ) ( ) ( ) ( ) ( ) ( So, ) ( ) ( ) ( a F b F dx x f b a - =
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The Fundamental Theorem of Calculus Part 1: If f if continuous on
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