#5 ,VECTOR ,9TEGRAL ,CALCULUS ,"P ;,I_4 ,TWO-,DIM5.NAL ,!ORY #5.1 ,,9TRODUC;N ,! TOPIC ( ? *APT] IS ..L9E & SURFACE .9TEGRALS4 ,X W 2 SE5 T ^! C BO? 2 REG>D$ Z 9TEGRALS ( VECTORS & T ! PR9CIPAL !OREMS C 2 MO/ SIMPLY /AT$ 9 T]MS ( VECTORS2 H;E ! TITLE 8VECTOR 9TEGRAL CALCULUS40 ,A FAMILI> L9E 9TEGRAL IS T ( >C L5G?3 "!%,C]DS_4 ,! SUBSCRIPT ;,C 9DICATES T "O IS M1SUR+ ! L5G? (A CURVE ;,C, Z 9 ,FIG4 #5.1_4 ,IF ;,C IS GIV5 9 P>AMETRIC =M X .K X(T), Y .K Y(T), ! L9E 9TEGRAL REDUCES 6! ORD9>Y DEF9ITE 9TEGRAL3 "!%,C]DS .K !;T;;1^T^;2 ">(?DX/DT#)^2"+(?DY/DT#)^2"]DT_4 ,IF ! CURVE ;,C REPRES5TS A WIRE ^: D5S;Y (MASS P] UNIT L5G?) V>IES AL;G ;,C, !N ! WIRE HAS A TOTAL A#BGI MASS ,M .K "!%,C]F(X, Y)DS, ": F(X, Y) IS ! D5S;Y AT ! PO9T (X, Y) (! WIRE4 ,! NEW 9TEGRAL C 2 EXPRESS$ 9 T]MS (A P>AMET] Z PREVI\SLY OR C 2 ?"\ ( SIMPLY Z A LIMIT (A SUM "!%,C]F(X, Y)DS .K LIM ".,S%I .K #1