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Unformatted text preview: MATH0201 BASIC CALCULUS MATH0201 BASIC CALCULUS Definite Integration Dr. WONG Chi Wing Department of Mathematics, HKU MATH0201 BASIC CALCULUS Riemann Sum and Definite Integral Evaluation of Definite Integrals Fundamental Theorem of Calculus Definite Integral by the Method of Substitution Applications of Definite Integral Area between Curves Volume of Solids of Revolution Applications in Economics Applications in Life and Social Sciences Reference § 5.3–5.6 of the textbook. MATH0201 BASIC CALCULUS Riemann Sum and Definite Integral Archimedes’ Quadrature of Parabola Archimedes (3rd century B.C.) found the area of a segment of a parabola by the method of exhaustion . I BM = CM and AM is parallel to the axis of parabola. I Repeat the process on every sub-segment of the parabola. It can be proved that Area ( AGB ) + Area ( AHC ) = 1 4 Area ( ABC ) : The area of the segment of the parabola is 4 3 Area ( ABC ) . MATH0201 BASIC CALCULUS Riemann Sum and Definite Integral Quadrature of Parabola in 17th Century Consider f ( x ) = x 2 over [ ; 1 ] . I Finer partition may yield a better approximation of the area under the curve. I If we partition [ ; 1 ] into n equal parts, the sum of the area of rectangles is ( n 1 ) n ( 2 n 1 ) 6 n 3 : MATH0201 BASIC CALCULUS Riemann Sum and Definite Integral Definition 2 (Riemann Sum (p.401)) Let f be a function and [ a ; b ] dom ( f ) . I Partition [ a ; b ] into n subintervals: [ x ; x 1 ] , [ x 1 ; x 2 ] , ::: , [ x n 1 ; x n ] where x = a and x n = b . I The length of the subintervals may not be the same. I Over each subinterval [ x k 1 ; x k ] , a rectangle of height f ( c k ) is erected where c k 2 [ x k 1 ; x k ] . The sum of the areas of these n rectangles is called a Riemann sum of f on [ a ; b ] . MATH0201 BASIC CALCULUS Riemann Sum and Definite Integral Theorem 3 (Limit of Riemann Sum) If f is continuous on [ a ; b ] , then the Riemann sums of f on [ a ; b ] will get closer and closer to a certain real number L by considering finer and finer partitions. Definition 4 (Definite Integral (p.401)) Let f be continuous on [ a ; b ] . The limit L of Riemann sums of f over [ a ; b ] is called the definite integral of f from a to b and it is denoted by Z b a f ( x ) dx : f ( x ) is the integrand , a is the lower limit of integration , and b is the upper limit of integration . MATH0201 BASIC CALCULUS Riemann Sum and Definite Integral Geometric Interpretation of Definite Integral (p.399) The definite integral Z b a f ( x ) dx represents the cumulative sum of the signed areas between the graph of f and the x –axis from x = a to x = b ....
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