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# Class Note for MATH 2433 with Professor Zhang at UH

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This 23 page Class Notes was uploaded by an elite notetaker on Friday February 6, 2015. The Class Notes belongs to a course at University of Houston taught by a professor in Fall. Since its upload, it has received 18 views.

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Date Created: 02/06/15

Sigma notations 251102i24182011o Sigma notations 251102i24182011o 2102jij 234345111213 393 211 2 1IJ 20i1i2i3 2 1 3i6 16560225 Sigma notations 251102i24182011o 2102jij 234345111213 393 2quot1 2 1 I J 0i1i2i3 2 103i6 I 1 16560225 im fn quot7 im fn 211 2115 2i1 21130 im jn kq im jn kq 2i1 2j12k1auk Zi1 211 211 3 Sigma notations im fn 7 im fn 211 Zj1aall 21 2115 Sigma notations im fn 7 im fn 211 Zj1aall 21 2115 2132quotzjiltaij by 2112quot253aa 213qquot2j32ba Sigma notations im fn 7 im fn 21 210cau 7 XXI1 2113 im fn 7 im fn im fn 21 Ej1aj by 7 21 2113 2i12j1bu ZETZEHENJJ 259quot30 qu Sigma notations 2T2 Xaij EETEEQ73U 2139quot213 ag by 2sziaa 2 qquot2 ba 2T2 aibj 253130 if y 254quot2 lti2sinltjgtgt 2131392 253sinm Double integrals Qfxydxdy where Q is a region in the xy plane fxy is a function of two variables Double integrals over a rectangle partitions Double integrals over a rectangle partitions Partition of X PXX0X1Xmwhereaxoltx1ltXmb Double integrals over a rectangle partitions Partition of X PX X0X1 Xmwhere a X0 lt X1 Partition of X Py yo7y1 7yn7where 6 yo lty1 ltXmb ltynd Double integrals over a rectangle partitions Partition ofx PX X0X1 Xmwhere ax0 ltX1ltXmb Partition ofx Py yoy1 yn7where cyo ltV1ltVnd Partition of Q Rij3Xi71 X XI7YI71 Y YI7Where1 i m71 SIS Double integrals over a rectangle definition gt P upper sum for f U P 2519quot2jIjMU area of R 2119quot2j3 MjAxij where M is the a maximum value off on R Double integrals over a rectangle definition gt P upper sum for f U P 2519quot2jIjMU area of R 2r392j3 MijAy where M is the a maximum value off on R gt P lower sum for f LP 27392j3 mjarea of R 2qnzjlg mijij where mg is the a minimum value of f on R Double integrals over a rectangle definition gt P upper sum for f U P 2519quot2jIjMU area of R 2r392j3 MijAy where M is the a maximum value off on R gt P lower sum for f LP 27392j3 mjarea of R 2qnzjlg mijij where mg is the a minimum value of f on R gt Definition Let f be continuous on a closed rectangle R The unique number I that satisfies the inequality LfP I UfPfor all partitions PofR is called the double integral of f over R and is denoted by Rfxydxdy Double integral as a volume gt If f is continuous and nonnegative on R then Rfxydxdy is the volume ofthe solid bounded below by R and above by the surface 2 fxy Double integral as a volume gt If f is continuous and nonnegative on R then Rfxydxdy is the volume ofthe solid bounded below by R and above by the surface 2 fxy gt If fxy 1 over area ofRfxydxdy R Properties of double integral gt Linearity f fQWfOCY BQX7YdXdV ocf f9fX7VdXdV BffggX7Ydxdy Properties of double integral gt Linearity f fQWfOCY BQX7YdXdV ocf f9fX7VdXdV BffggX7Ydxdy gt Order if f 2 O on Q then ff9fxydxdy 2 O Properties of double integral gt Linearity f fQWfOCY BQX7YdXdV ocf f9fX7VdXdV BffggX7Ydxdy gt Order if f 2 O on Q then ff9fxydxdy 2 0 if f 2 g on Q then ff9fxydxdy 2 ff gxydxdy Properties of double integral gt Linearity f fQWfOCY BQX7YdXdV ocf f9fX7VdXdV BffggX7Ydxdy gt Order if f 2 O on Q then ff9fxydxdy 2 0 if f 2 g on Q then ff9fxydxdy 2 ff gxydxdy gt Additivity If 9 is broken into a finite number of nonoverlapping basic regions 2192 9 then AlfxydxdyA1fxydxdyA fxydxdy Meanvalue theorem for the double integrals gt If f is continuous on 9 there is a point X0y0 in Q for which Qfxydxdy fx0y0 area ofR fx0y0 is called the average value of f on Q Meanvalue theorem for the double integrals gt If f is continuous on 9 there is a point X0y0 in Q for which Qfxydxdy fx0y0 area ofR fx0y0 is called the average value of f on 9 gt If both f and g are continuous on Q and g is nonnegative on Q then there exist a point X0y0 in Q for which AlfX7Y9X7VdXdVfX07YOA19X7VdXd fx0y0 is called the gweighted average value of f on Q

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