Area and the Definite Integral Michael Freeze MAT 151 UNC - PowerPoint PPT Presentation

Area and the Definite Integral Michael Freeze MAT 151 UNC Wilmington Summer 2013 1 / 6

Section 7.3 :: Area and the Definite Integral 2 / 6

Archimedes’ Quadrature of the Parabola • Archimedes was able to show that the area of a parabolic section is 4 3 times the area of an inscribed triangle. • A key component of his method was the summation of a geometric series. • The method did not extend to the quadrature of the circle. 3 / 6

The Definite Integral � b We write a f ( x ) dx to denote the signed area between the graph of y = f ( x ) and the x -axis over the interval a ≤ x ≤ b . The value of the definite integral is given by � n � b � � f ( x ) dx = lim f ( x i ) ∆ x , n →∞ a i =1 provided the limit exists, where ∆ x = b − a n and x i is any value of x in the i th subinterval of [ a , b ]. 4 / 6

Total Change in F ( x ) If f ( x ) gives the rate of change of F ( x ) for x in [ a , b ], then the total change in F ( x ) as x goes from a to b is given by � n � b � � lim f ( x i ) ∆ x = f ( x ) dx . n →∞ a i =1 5 / 6

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