8 03 at ESG Supplemental Notes Special for 8 044 Spring 2006 We have already seen and used Leibnitz s Formula for di erentiating an integral in the form d dy xR y xL y dxR dxL f x y dx f xR y f xL y dy dy xR y xL y f x y dx y The above notation is similar to that used in the 18 023 text the subscripts for xL y andd xR y are for Left and Right corresponding to the appearance of the graphs of these functions in the x y plane The formula can be extended in many ways the limits could be functions of several variables in which case partial derivatives would be used throughout Or for the case of multiple integrals to nd probably distributions the formula might be applied successively to nested integrals Consider for instance P z xR z xL z dx yT x z yB x z f x y z dy Here the subscripts on yB x z and yT x z are for Bottom and Top again corresponding to the appearance of the graphs of these functions in the x y plane but di erent graphs for di erent values of the independent variable z Using Liebnitz s formula dP z dxR yT xR z dxL yT xL z f xR y z dy f xL y z dy dz dz yB xR z dz yB xL z xR z yT x z dx f x y z dy z yB x z xL z The partial derivative in the last term is now evaluated as z yT x z yB x z yT yB f x yT x z z f x yB x z z z z yT x z f x y z dy yB x z z f x y z dy 1 Whew Altogether we have ve terms dxR dP z dz dz yT xR z yB xR z f xR y z dy dxL yT xL z f xL y z dy dz yB xL z xR z yT f x yT x z z dx z xL z xR z yB f x yB x z z dx z xL z yT x z xR z dx f x y z dy xL z yB x z z The above is the full blown form and those with masochistic streaks may continue the process to any number of nested integrals However for current 8 044 purposes it s best not to get lost in the details That is take another look at the orignal double integral P z xR z xL z dx yT x z yB x z f x y z dy The independent variable z appears ve times in this expression and so the derivative has ve terms We can summarize the process by saying If z appears in the limit of an integral use the Fundamental Theorem of Calculus to di erentiate the integral multiplying by the derivative of the limit with the appropriate sign and evaluating the integrand at that limit If z appears in an integrand take the partial derivative of the integrand with respect to z Consider for example the situation of Problem 3 from Problem Set 3 In this case P z z is the area can be written in a form such that xL yB and f are constants with the result that xR z dxR yT xR z yT d P z f dx dy dz dz yB z xL So a little bit of e ort to show a useful technique results in a much easier calculation for those of us who still can t integrate by parts reliably 2