15.4 Definite Integrals and the Fundamental Theorem of Calculus.pdf

15.4 Definite Integrals and the Fundamental Theorem of Calculus.pdf

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15.4 Definite Integrals and the Fundamental Theorem of Calculus If ) ( x f is continuous on [ a , b ], then: ) ( ) ( ) ( a F b F dx x f b a = where F is the general antiderivative of f (note that there is no arbitrary constant C added to definite integrals). This formula is known as the (1 st ) Fundamental Theorem of Calculus. From a graphical standpoint, this integral represents the area of the region below ) ( x f but above the x -axis if 0 ) ( x f for all x on [ a , b ] or -1 times the area of the region above ) ( x f but below the x -axis if 0 ) ( x f for all x on [ a , b ]. dx x f b a ) ( A = dx x f b a ) ( A =
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