5.3 The Definite Integral & the Fundamental Theorem of CalculusMAC2233 (3.4) FIU, MDC-North5.3 The Definite Integral & the Fundamental Theorem of CalculusArea Under a Curve: If f(x) is continuous and f(x) ≥ 0 on the interval a ≤ x ≤ b, then the region under the curve y = f(x) over the interval a ≤ x ≤ b has area( ) ( )( )nnA f x f x f x x1 2lim ...��� �= + + + D� � where xj is the left endpoint of the jth subinterval if the interval a ≤ x ≤ b is divided into n equal parts, each of length b axn-D =. This area is more easily computed using the fundamental theorem of calculus: If f(x) is on the interval a ≤ x ≤b, then ( ) ( ) ( ) ( )bbaaf x dx F x F b F a= = -�. Rules for Definite Integrals: If f and g are any continuous function on a ≤ x ≤ b, then [1] Constant Multiple Rule: b ba akf x k f x( ) ( )=� � for constant k; [2] Sum or Difference Rule: ( )b b ba a af x g x dx f x dx g x dx( ) ( ) ( )� �� = �� �� � �; [3] Zero Rule: aaf x dx( ) 0=�; [4] Inverse Rule: a bb af x dx f x dx( ) ( )=-� �; [5] Subdivision Rule:b b ca c af x dx f x dx f x dx( ) ( ) ( )= +� � �.Evaluate the given integral using the fundamental theorem of calculus:1.dx12p-� 2.u du412�3.( )t te e dtln 20--�( ) ( )( )t te e e e e e eln 21lnln 2 ln 2 0 0201 12 1 1 2 22 2- -+ + - + + - + + -= = = =4.( )xdxx222131+�5.eedxx x21ln�MAC2233 (3.4) FIU, MDC-North6. ( )x x x dx05 213 3 2 5-- - + +�If f(x) and g(x) are continuous on the interval -3 ≤ x ≤ 2 and satisfy( ) ( ) ( ) ( )f x dx g x dx f x dx g x dx2 2 1 13 3 3 35 2 0 4- - - -= =- = =� � � �,evaluate the given integral using this information along with rules for definite integrals:7.( )g x dx21�8.( ) ( )f x g x dx132 3-� �+� ��9. Find the area of the region R that lies under the curve y = ( )x x 1+, over 0 ≤ x ≤
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