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# Calculus III MATH 2433

UH

GPA 3.69

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This 23 page Class Notes was uploaded by Alvena McDermott on Saturday September 19, 2015. The Class Notes belongs to MATH 2433 at University of Houston taught by Zhigang Zhang in Fall. Since its upload, it has received 23 views. For similar materials see /class/208366/math-2433-university-of-houston in Mathmatics at University of Houston.

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Date Created: 09/19/15

Sigma notations 251102i24182011o Sigma notations 251102i24182011o 2102jij 234345111213 393 211 2 1IJ 20i1i2i3 2 1 3i6 16560225 Sigma notations 251102i24182011o 2 1 2j3i1 234345111213 393 2quot1 2 1 I J 0i1i2i3 1 lt3i6gt 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 ax0 ltX1ltXmb Partition of X PyVO7Y17 7Yn7WhereCV0ltV1 ltVnd 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 2391qquot2j3 Mj 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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